A Visual Introduction to Differential Forms and Calculus on Manifolds

470 Pages·2019·15.235 MB·English

by  Fortney, J.P.

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
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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