Fortney, J.P. - A Visual Introduction to Differential Forms and Calculus on Manifolds (2019, Springer Internatio.pdf

Fortney, J.P. - A Visual Introduction to Differential Forms and Calculus on Manifolds (2019, Springer Internatio.pdf

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AVisualIntroductiontoDifferentialFormsandCalculusonManifoldsJonPierreFortneyAVisualIntroductiontoDifferentialFormsandCalculusonManifoldsJonPierreFortneyAVisualIntroductiontoDifferentialFormsandCalculusonManifoldsJonPierreFortneyDepartmentofMathematicsandStatisticsZayedUniversityDubai,UnitedArabEmiratesISBN978-3-319-96991-6ISBN978-3-319-96992-3（eBook）https:

/doi.org/10.1007/978-3-319-96992-3LibraryofCongressControlNumber:

2018952359MathematicsSubjectClassification（2010）:

53-01,57-01,58-01,53A45,58C20,58C35SpringerNatureSwitzerlandAG2018Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped.Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.ThisbookispublishedundertheimprintBirkhuser,www.birkhauser-bytheregisteredcompanySpringerNatureSwitzerlandAGTheregisteredcompanyaddressis:

Gewerbestrasse11,6330Cham,SwitzerlandTomyparents,DanielandMarleneFortney,foralloftheirloveandsupport.PrefaceDifferentialforms,whilenotquiteubiquitousinmathematics,arecertainlycommon.Andtheroledifferentialformsplayappearsinawiderangeofmathematicalfieldsandapplications.Differentialforms,andtheirintegrationonmanifolds,arepartofthefoundationalmaterialwithwhichitisnecessarytobeproficientinordertotackleawiderangeofadvancedtopicsinbothmathematicsandphysics.Someupper-levelundergraduatebooksandnumerousgraduatebookscontainachapterondifferentialforms,butgenerallytheintentofthesechaptersistoprovidethecomputationaltoolsnecessaryfortherestofthebook,nottoaidstudentsinactuallyobtainingaclearunderstandingofdifferentialformsthemselves.Furthermore,differentialformsoftendonotshowupinthetypicallyrequiredundergraduatemathematicsorphysicscurriculums,makingitbothunlikelyanddifficultforstudentstogainadeepandintuitivefeelingforthem.Oneofthetwoaimsofthisbookistoaddressandremedyexactlythisgapinthetypicalundergraduatemathematicsandphysicscurriculums.Additionally,itisduringthesecondyearandthirdyearthatundergraduatemathematicsmajorsaremakingthetransitionfromtheconcretecomputation-basedsubjectsgenerallyfoundinhighschoolandlower-levelundergraduatecoursestotheabstracttopicsgenerallyfoundinupper-levelundergraduateandgraduatecourses.Thisisatrickyandchallengingtimeformanyundergraduatestudents,anditisduringthisperiodthatmostundergraduateprogramsseethehighestattritionrates.Furthermore,whilemanyundergraduatemathematicsprogramsrequiremathematicalstructuresorintroductiontoproofsclass,therearealsomanyprogramsdonot.Andoftenasinglecoursemeanttohelpstudentstransitionfromtheconcretecomputationsofcalculustotheabstractnotionsoftheoreticalmathematicsisnotenough;amajorityofstudentsneedmoresupportinmakingthistransition.Thesecondaimofthisbookhasbeentohelpstudentsmakethistransitiontoamathematicallymoreabstractandmaturewayofthinking.Thus,theintendedaudienceforthisbookisquitebroad.Fromtheperspectiveofthetopicscovered,thisbookwouldbecompletelyappropriateforamoderngeometrycourse;inparticular,acoursethatismeanttohelpstudentsmakethejumpfromEuclidian/Hyperbolic/Ellipticgeometrytodifferentialgeometry,oritcouldbeusedinthefirstsemesterofatwo-semestersequenceindifferentialgeometry.Itwouldalsobeappropriateasanadvancedcalculuscoursethatismeanttohelpstudentstransitiontocalculusandanalysisonmanifolds.Additionally,itwouldbeappropriateforanundergraduatephysicsprogram,particularlyonewithamoretheoreticalbent,orinaphysicshonorsprogram;itcouldbeusedinageometryforphysicscourse.Finally,fromthisperspective,itisalsoaperfectreferenceforgraduatestudentsenteringanyfieldwhereathoroughknowledgeofdifferentialformsisnecessaryandwhofindtheylackthenecessarybackground.Thoughgraduatestudentsarenottheintendedaudience,theycouldreadandassimilatetheideasquitequickly,therebyenablingthemtogainafairlydeepinsightintothebasicnatureofdifferentialformsbeforetacklingmoreadvancedmaterial.Butfromtheperspectiveofhelpingundergraduatestudentsmakethetransitiontoabstractmathematics,thisbookisabsolutelyappropriateforanyandallsecond-orthird-yearundergraduatemathematicsmajors.Itsmathematicalprerequisitesarelight;acourseinvectorcalculusiscompletelysufficientandthefewnecessarytopicsinlinearalgebraarecoveredintheintroductorychapter.However,thisbookhasbeencarefullywrittentoprovideundergraduatesthescaffoldingnecessarytoaidtheminthetransitiontoabstractmathematics.Infact,thismaterialdove-tailswithvectorcalculus,withwhichstudentsarealreadyfamiliar,makingitaperfectsettingtohelpstudentstransitiontoadvancedtopicsandabstractwaysofthinking.Thusthisbookwouldbeidealinasecond-orthird-yearcoursewhoseintentistoaidstudentsintransitioningtoupper-levelmathematicscourses.Assuch,Ihaveemployedanumberofdifferentpedagogicalapproachesthataremeanttocomplementeachotherandprovideagradualyetrobustintroductiontobothdifferentialformsinparticularandabstractmathematicsingeneral.First,Ihavemadeagreatdealofefforttograduallybuilduptothebasicideasandconcepts,sothatdefinitions,whenmade,donotappearoutofnowhere;Ihavespentmoretimeexploringthe“how”and“why”ofthingsthanistypicalformostpost-calculusmathbooks.Additionally,thetwomajorproofsthataredoneinthisbook（thegeneralizedStokestheoremandthePoincarviiviiiPrefacelemma）aredoneveryslowlyandcarefully,providingmoredetailthanisusual.Second,thisbooktriestoexplainandhelpthereaderdevelop,asmuchaspossible,theirgeometricintuitionasitrelatestodifferentialforms.Toaidinthisendeavorthereareover250figuresinthebook.Theseimagesplayacrucialroleinaidingthestudenttounderstandandvisualizetheconceptsbeingdiscussedandareanintegralpartoftheexposition.Third,Studentsbenefitfromseeingthesameideapresentedandexplainedmultipletimesandfromdifferentperspectives;therepetitionaidsinlearningandinternalizingtheidea.AnumberofthemoreimportanttopicsarediscussedinboththeRnsettingaswellasinthesettingofmoreabstractmanifolds.Also,manytopicsarediscussedfromavisual/geometricapproachaswellasfromacomputationalapproach.Finally,thereareover200exercisesinterspersedwiththetextandabout200additionalend-of-chapterexercises.Theendofchapterquestionsareprimarilycomputational,meanttohelpstudentsgainfamiliarityandproficiencywiththenotationandconcepts.Questionsinterspersedinthetextrangefromtrivialtochallengingandaremeanttohelpstudentsgenuinelyengagewiththereadings,absorbfundamentalideas,andlookcarefullyandcriticallyatvariousstepsofthecomputationsdoneinthetext.Takentogether,thesequestionswillnotonlyallowstudentstogainadeeperunderstandingofthematerial,butalsogainconfidenceintheirabilitiesandinternalizetheessentialnotationandideas.Puttingallofthesepedagogicalstrategiestogethermayresultinanexpositionthat,toanexpert,wouldseemattimestobeunnecessarilylong,butthisbookisbasedonmyownexperiencesandreflectionsinbothlearningandteachingandisentirelywrittenwithstudentsfairlynewtomathematicsinmind.Iwantmyreaderstotrulyunderstandandinternalizetheseideas,togainadeeperandmoreaccurateperceptionofmathematics,andtoseethebeautifulinterconnectednessofthesubject;Iwantmyreaderstowalkawayfeelingthattheyhavegenuinelymasteredabodyofknowledgeandnotsimplylearnedasetofdisconnectedfacts.Coveringthefullbookisprobablytoomuchtoaskofmoststudentsinaone-semestercourse,butthereareanumberofdifferentpathwaysthroughthebookbasedontheoverallemphasisoftheclass.ProvidedbelowaretheonesIconsidermostappropriate:

1.Forschoolsonthequartersystemorforaseminarclass:

1（optional）,2,3,4,6,7,9（optional）.2.Emphasizingdifferentialformsandgeometry:

1（optional）,29,10（optional）,11,AppendixB12（optional）.3.Emphasizingphysics:

1（optional）,27,9,11,12,AppendixA（optional）,AppendixB35（optional）.4.Emphasizingthetransitiontoabstractmathematics:

1（optional）,24,611.5.Advancedstudentsorasafirstcoursetoanupper-levelsequenceindifferentialgeometry:

1（optional）,211,AppendixA（optional）,AppendixB（optional）.Awordofwarning,AppendixAontensorswasincludedinordertoprovidetheproofoftheglobalformulaforexteriordifferentiation,aproofIfeltwasessentialtoprovideinthisbookandwhichreliesontheliederivative.However,fromapedagogicalperspecti