Jon Pierre Fortney A Visual Introduction to Differential Forms and Calculus on Manifolds A Visual Introduction to Differential Forms and Calculus on Manifolds Jon Pierre Fortney A Visual Introduction to Differential Forms and Calculus on Manifolds JonPierreFortney DepartmentofMathematicsandStatistics ZayedUniversity Dubai,UnitedArabEmirates ISBN978-3-319-96991-6 ISBN978-3-319-96992-3 (eBook) https://doi.org/10.1007/978-3-319-96992-3 LibraryofCongressControlNumber:2018952359 MathematicsSubjectClassification(2010):53-01,57-01,58-01,53A45,58C20,58C35 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialisconcerned,specificallytherights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction onmicrofilms orinany other physical way, and transmission or informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecific statement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedate ofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorfor anyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutional affiliations. ThisbookispublishedundertheimprintBirkhäuser,www.birkhauser-science.combytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To myparents,Danieland MarleneFortney,forall oftheirloveandsupport. Preface Differentialforms,while notquite ubiquitousin mathematics,are certainlycommon.And the role differentialformsplay appearsinawiderangeofmathematicalfieldsandapplications.Differentialforms,andtheirintegrationonmanifolds,are partofthefoundationalmaterialwithwhichitisnecessarytobeproficientinordertotackleawiderangeofadvancedtopics in both mathematics and physics. Some upper-levelundergraduatebooksand numerousgraduatebookscontain a chapter ondifferentialforms,butgenerallytheintentofthese chaptersistoprovidethecomputationaltoolsnecessaryforthe rest of the book,notto aid studentsin actually obtaininga clear understandingof differentialformsthemselves. Furthermore, differentialformsoftendonotshowupinthetypicallyrequiredundergraduatemathematicsorphysicscurriculums,making itbothunlikelyanddifficultforstudentstogainadeepandintuitivefeelingforthem.Oneofthetwoaimsofthisbookisto addressandremedyexactlythisgapinthetypicalundergraduatemathematicsandphysicscurriculums. Additionally,itisduringthesecondyearandthirdyearthatundergraduatemathematicsmajorsaremakingthetransition fromtheconcretecomputation-basedsubjectsgenerallyfoundinhighschoolandlower-levelundergraduatecoursestothe abstract topics generally found in upper-level undergraduate and graduate courses. This is a tricky and challenging time for many undergraduate students, and it is during this period that most undergraduate programs see the highest attrition rates. Furthermore, while many undergraduate mathematics programs require mathematical structures or introduction to proofs class, there are also many programsdo not. And often a single course meant to help students’ transition from the concrete computationsof calculusto the abstractnotionsoftheoreticalmathematicsis notenough;a majorityof students needmoresupportinmakingthistransition.Thesecondaimofthisbookhasbeentohelpstudentsmakethistransitiontoa mathematicallymoreabstractandmaturewayofthinking. Thus, the intended audience for this book is quite broad. From the perspective of the topics covered, this book would be completely appropriate for a modern geometry course; in particular, a course that is meant to help students make the jumpfromEuclidian/Hyperbolic/Ellipticgeometrytodifferentialgeometry,oritcouldbeusedinthefirstsemesterofatwo- semestersequenceindifferentialgeometry.Itwouldalsobeappropriateasanadvancedcalculuscoursethatismeanttohelp students’transitiontocalculusandanalysisonmanifolds.Additionally,itwouldbeappropriateforanundergraduatephysics program,particularlyonewitha moretheoreticalbent,orina physicshonorsprogram;itcouldbeusedina geometryfor physicscourse.Finally,fromthisperspective,itisalsoaperfectreferenceforgraduatestudentsenteringanyfieldwherea thoroughknowledgeofdifferentialformsisnecessaryandwhofindtheylackthenecessarybackground.Thoughgraduate studentsarenottheintendedaudience,theycouldreadandassimilatetheideasquitequickly,therebyenablingthemtogain afairlydeepinsightintothebasicnatureofdifferentialformsbeforetacklingmoreadvancedmaterial. But from the perspective of helping undergraduate students make the transition to abstract mathematics, this book is absolutelyappropriateforanyandallsecond-orthird-yearundergraduatemathematicsmajors.Itsmathematicalprerequisites arelight;acourseinvectorcalculusiscompletelysufficientandthefewnecessarytopicsinlinearalgebraarecoveredinthe introductorychapter.However,thisbookhasbeencarefullywrittentoprovideundergraduatesthescaffoldingnecessaryto aidtheminthetransitiontoabstractmathematics.Infact,thismaterialdove-tailswithvectorcalculus,withwhichstudents arealreadyfamiliar,makingitaperfectsettingtohelpstudentstransitiontoadvancedtopicsandabstractwaysofthinking. Thusthisbookwouldbeidealinasecond-orthird-yearcoursewhoseintentistoaidstudentsintransitioningtoupper-level mathematicscourses. As such, I have employeda numberof differentpedagogicalapproachesthat are meantto complementeach other and provideagradualyetrobustintroductiontobothdifferentialformsinparticularandabstractmathematicsingeneral.First,I havemadeagreatdealofefforttograduallybuilduptothebasicideasandconcepts,sothatdefinitions,whenmade,donot appearoutofnowhere;Ihavespentmoretimeexploringthe“how”and“why”ofthingsthanistypicalformostpost-calculus mathbooks.Additionally,thetwomajorproofsthataredoneinthisbook(thegeneralizedStokes’theoremandthePoincaré vii viii Preface lemma)aredoneveryslowlyandcarefully,providingmoredetailthanisusual.Second,thisbooktriestoexplainandhelp thereaderdevelop,asmuchaspossible,theirgeometricintuitionasitrelatestodifferentialforms.Toaidinthisendeavor there are over 250 figures in the book. These images play a crucial role in aiding the student to understandand visualize the conceptsbeing discussed and are an integralpart of the exposition.Third, Studentsbenefitfromseeing the same idea presentedandexplainedmultipletimesandfromdifferentperspectives;therepetitionaidsinlearningandinternalizingthe idea.AnumberofthemoreimportanttopicsarediscussedinboththeRn settingaswellasinthesettingofmoreabstract manifolds. Also, many topics are discussed from a visual/geometric approach as well as from a computationalapproach. Finally,thereareover200exercisesinterspersedwith thetextandabout200additionalend-of-chapterexercises.The end ofchapterquestionsareprimarilycomputational,meanttohelpstudentsgainfamiliarityandproficiencywiththenotation andconcepts.Questionsinterspersedinthetextrangefromtrivialtochallengingandaremeanttohelpstudentsgenuinely engagewith the readings,absorbfundamentalideas, andlookcarefullyandcritically atvarioussteps ofthe computations doneinthetext.Takentogether,thesequestionswillnotonlyallowstudentstogainadeeperunderstandingofthematerial, butalsogainconfidenceintheirabilitiesandinternalizetheessentialnotationandideas. Putting all of these pedagogicalstrategies together may result in an expositionthat, to an expert, would seem at times to be unnecessarilylong, butthis bookis based on my own experiencesand reflectionsin bothlearningand teaching and is entirely written with students fairly new to mathematicsin mind. I wantmy readersto truly understandand internalize these ideas, to gaina deeperand more accurateperceptionof mathematics,and to see the beautifulinterconnectednessof thesubject;Iwantmyreaderstowalkawayfeelingthattheyhavegenuinelymasteredabodyofknowledgeandnotsimply learnedasetofdisconnectedfacts. Coveringthefullbookisprobablytoomuchtoaskofmoststudentsinaone-semestercourse,butthereareanumberof differentpathwaysthroughthebookbasedontheoverallemphasisoftheclass.ProvidedbelowaretheonesIconsidermost appropriate: 1. Forschoolsonthequartersystemorforaseminarclass:1(optional),2,3,4,6,7,9(optional). 2. Emphasizingdifferentialformsandgeometry:1(optional),2–9,10(optional),11,AppendixB1–2(optional). 3. Emphasizingphysics:1(optional),2–7,9,11,12,AppendixA(optional),AppendixB3–5(optional). 4. Emphasizingthetransitiontoabstractmathematics:1(optional),2–4,6–11. 5. Advancedstudentsorasafirstcoursetoanupper-levelsequenceindifferentialgeometry:1(optional),2–11,Appendix A(optional),AppendixB(optional). A word of warning, AppendixA on tensors was included in order to providethe proof of the global formula for exterior differentiation, a proof I felt was essential to provide in this book and which relies on the lie derivative. However, from a pedagogical perspective Appendix A is probably too terse and lacks the necessary examples to be used as a general introductiontotensors,atleastifonewishestocoveranythingbeyondthemeredefinitionsandbasicidentities.Instructors shouldkeepthisinmindwhendecidingwhetherornottoincorporatethisappendixintotheirclasses. Finally, I would like to express my sincere appreciation to Ahmed Matar and Ron Noval for all of their invaluable comments and suggestions. Additionally, I would like to thank my friend Rene Hinojosa for his ongoing encouragement andsupport. Dubai,UnitedArabEmirates JonPierreFortney May2018 Contents 1 BackgroundMaterial ................................................................................................... 1 1.1 ReviewofVectorSpaces.......................................................................................... 1 1.2 VolumeandDeterminants ........................................................................................ 16 1.3 DerivativesofMultivariableFunctions........................................................................... 23 1.4 Summary,References,andProblems............................................................................. 27 1.4.1 Summary................................................................................................. 27 1.4.2 ReferencesandFurtherReading........................................................................ 28 1.4.3 Problems................................................................................................. 28 2 AnIntroductiontoDifferentialForms................................................................................ 31 2.1 CoordinateFunctions ............................................................................................. 31 2.2 TangentSpacesandVectorFields................................................................................ 37 2.3 DirectionalDerivatives............................................................................................ 43 2.4 DifferentialOne-Forms........................................................................................... 53 2.5 Summary,References,andProblems............................................................................. 65 2.5.1 Summary................................................................................................. 65 2.5.2 ReferencesandFurtherReading........................................................................ 66 2.5.3 Problems................................................................................................. 66 3 TheWedgeproduct ...................................................................................................... 69 3.1 AreaandVolumewiththeWedgeproduct........................................................................ 69 3.2 GeneralTwo-FormsandThree-Forms ........................................................................... 82 3.3 TheWedgeproductofn-Forms................................................................................... 88 3.3.1 AlgebraicProperties..................................................................................... 88 3.3.2 SimplifyingNotation.................................................................................... 90 3.3.3 TheGeneralFormula.................................................................................... 93 3.4 TheInteriorProduct............................................................................................... 97 3.5 Summary,References,andProblems............................................................................. 100 3.5.1 Summary................................................................................................. 100 3.5.2 ReferencesandFurtherReading........................................................................ 102 3.5.3 Problems................................................................................................. 102 4 ExteriorDifferentiation................................................................................................. 107 4.1 AnOverviewoftheExteriorDerivative ......................................................................... 107 4.2 TheLocalFormula................................................................................................ 109 4.3 TheAxiomsofExteriorDifferentiation.......................................................................... 112 4.4 TheGlobalFormula............................................................................................... 114 4.4.1 ExteriorDifferentiationwithConstantVectorFields.................................................. 114 4.4.2 ExteriorDifferentiationwithNon-ConstantVectorFields ............................................ 121 4.5 AnotherGeometricViewpoint.................................................................................... 130 4.6 ExteriorDifferentiationExamples................................................................................ 142 4.7 Summary,References,andProblems............................................................................. 147 ix x Contents 4.7.1 Summary................................................................................................. 147 4.7.2 ReferencesandFurtherReading........................................................................ 148 4.7.3 Problems................................................................................................. 149 5 VisualizingOne-,Two-,andThree-Forms............................................................................ 151 5.1 One-andTwo-FormsinR2....................................................................................... 151 5.2 One-FormsinR3 .................................................................................................. 160 5.3 Two-FormsinR3.................................................................................................. 166 5.4 Three-FormsinR3 ................................................................................................ 175 5.5 PicturesofFormsonManifolds.................................................................................. 175 5.6 AVisualIntroductiontotheHodgeStarOperator............................................................... 179 5.7 Summary,References,andProblems............................................................................. 186 5.7.1 Summary................................................................................................. 186 5.7.2 ReferencesandFurtherReading........................................................................ 187 5.7.3 Problems................................................................................................. 187 6 Push-ForwardsandPull-Backs........................................................................................ 189 6.1 CoordinateChange:ALinearExample.......................................................................... 189 6.2 Push-ForwardsofVectors......................................................................................... 196 6.3 Pull-BacksofVolumeForms..................................................................................... 201 6.4 PolarCoordinates ................................................................................................. 206 6.5 CylindricalandSphericalCoordinates........................................................................... 213 6.6 Pull-BacksofDifferentialForms................................................................................. 217 6.7 SomeUsefulIdentities............................................................................................ 223 6.8 Summary,References,andProblems............................................................................. 226 6.8.1 Summary................................................................................................. 226 6.8.2 ReferencesandFurtherReading........................................................................ 227 6.8.3 Problems................................................................................................. 227 7 ChangesofVariablesandIntegrationofForms ..................................................................... 229 7.1 IntegrationofDifferentialForms................................................................................. 229 7.2 ASimpleExample ................................................................................................ 235 7.3 Polar,Cylindrical,andSphericalCoordinates................................................................... 240 7.3.1 PolarCoordinatesExample............................................................................. 240 7.3.2 CylindricalCoordinatesExample....................................................................... 243 7.3.3 SphericalCoordinatesExample......................................................................... 244 7.4 IntegrationofDifferentialFormsonParameterizedSurfaces................................................... 245 7.4.1 LineIntegrals............................................................................................ 246 7.4.2 SurfaceIntegrals......................................................................................... 251 7.5 Summary,References,andProblems............................................................................. 254 7.5.1 Summary................................................................................................. 254 7.5.2 ReferencesandFurtherReading........................................................................ 255 7.5.3 Problems................................................................................................. 255 8 PoincaréLemma......................................................................................................... 259 8.1 IntroductiontothePoincaréLemma............................................................................. 259 8.2 TheBaseCaseandaSimpleExampleCase..................................................................... 261 8.3 TheGeneralCase.................................................................................................. 268 8.4 Summary,References,andProblems............................................................................. 275 8.4.1 Summary................................................................................................. 275 8.4.2 ReferencesandFurtherReading........................................................................ 275 8.4.3 Problems................................................................................................. 275 9 VectorCalculusandDifferentialForms .............................................................................. 277 9.1 Divergence......................................................................................................... 277 9.2 Curl ................................................................................................................ 284 9.3 Gradient............................................................................................................ 293 Contents xi 9.4 UpperandLowerIndices,Sharps,andFlats..................................................................... 294 9.5 RelationshiptoDifferentialForms ............................................................................... 298 9.5.1 Grad,Curl,DivandExteriorDifferentiation........................................................... 298 9.5.2 FundamentalTheoremofLineIntegrals ............................................................... 302 9.5.3 VectorCalculusStokes’Theorem ...................................................................... 303 9.5.4 DivergenceTheorem .................................................................................... 304 9.6 Summary,References,andProblems............................................................................. 305 9.6.1 Summary................................................................................................. 305 9.6.2 ReferencesandFurtherReading........................................................................ 306 9.6.3 Problems................................................................................................. 307 10 ManifoldsandFormsonManifolds ................................................................................... 309 10.1 DefinitionofaManifold.......................................................................................... 309 10.2 TangentSpaceofaManifold ..................................................................................... 313 10.3 Push-ForwardsandPull-BacksonManifolds.................................................................... 323 10.4 CalculusonManifolds............................................................................................ 326 10.4.1 DifferentiationonManifolds............................................................................ 327 10.4.2 IntegrationonManifolds................................................................................ 328 10.5 Summary,References,andProblems............................................................................. 332 10.5.1 Summary................................................................................................. 332 10.5.2 ReferencesandFurtherReading........................................................................ 334 10.5.3 Problems................................................................................................. 334 11 GeneralizedStokes’Theorem.......................................................................................... 337 11.1 TheUnitCubeIk.................................................................................................. 337 11.2 TheBaseCase:Stokes’TheoremonIk.......................................................................... 353 11.3 ManifoldsParameterizedbyIk................................................................................... 358 11.4 Stokes’TheoremonChains....................................................................................... 359 11.5 ExtendingtheParameterizations ................................................................................. 362 11.6 VisualizingStokes’Theorem..................................................................................... 363 11.7 Summary,References,andProblems............................................................................. 366 11.7.1 Summary................................................................................................. 366 11.7.2 ReferencesandFurtherReading........................................................................ 366 11.7.3 Problems................................................................................................. 366 12 AnExample:Electromagnetism....................................................................................... 369 12.1 Gauss’sLawsforElectricandMagneticFields.................................................................. 369 12.2 Faraday’sLawandtheAmpère-MaxwellLaw .................................................................. 375 12.3 SpecialRelativityandHodgeDuals.............................................................................. 380 12.4 DifferentialFormsFormulation .................................................................................. 384 12.5 Summary,References,andProblems............................................................................. 390 12.5.1 Summary................................................................................................. 390 12.5.2 ReferencesandFurtherReading........................................................................ 392 12.5.3 Problems................................................................................................. 392 A IntroductiontoTensors................................................................................................. 395 A.1 AnOverviewofTensors.......................................................................................... 395 A.2 RankOneTensors................................................................................................. 396 A.3 Rank-TwoTensors ................................................................................................ 404 A.4 GeneralTensors ................................................................................................... 407 A.5 DifferentialFormsasSkew-SymmetricTensors................................................................. 409 A.6 TheMetricTensor................................................................................................. 411 A.7 LieDerivativesofTensorFields.................................................................................. 414 A.8 SummaryandReferences......................................................................................... 431 A.8.1 Summary................................................................................................. 431 A.8.2 ReferencesandFurtherReading........................................................................ 434