Introduction to Tensor Analysis and the Calculus of Moving Surfaces

303 Pages·2013·2.447 MB·English

by  Pavel Grinfeld

Pavel Grinfeld
Introduction to
Tensor Analysis
and the Calculus
of Moving
Surfaces
Introduction to Tensor Analysis
and the Calculus of Moving Surfaces

Pavel Grinfeld
Introduction to Tensor
Analysis and the Calculus
of Moving Surfaces
123
PavelGrinfeld
DepartmentofMathematics
DrexelUniversity
Philadelphia,PA,USA
ISBN978-1-4614-7866-9 ISBN978-1-4614-7867-6(eBook)
DOI10.1007/978-1-4614-7867-6
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013947474
Mathematics Subject Classifications (2010): 4901, 11C20, 15A69, 35R37, 58A05, 51N20, 51M05,
53A05,53A04
©SpringerScience+BusinessMediaNewYork2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
The purpose of this book is to empower the reader with a magnificent new
perspectiveonawiderangeoffundamentaltopicsinmathematics.Tensorcalculus
isalanguagewithauniqueabilitytoexpressmathematicalideaswithutmostutility,
transparency, and elegance. It can help students from all technical fields see their
respectivefieldsinanewandexcitingway.Ifcalculusandlinearalgebraarecentral
tothereader’sscientificendeavors,tensorcalculusisindispensable.Thisparticular
textbookismeantforadvancedundergraduateandgraduateaudiences.Itenvisions
atimewhentensorcalculus,oncechampionedbyEinstein,isonceagainacommon
languageamongscientists.
A plethora of older textbooks exist on the subject. This book is distinguished
from its peers by the thoroughness with which the underlying essential elements
aretreated.Itfocusesagreatdealonthegeometricfundamentals,themechanicsof
changeofvariables,theproperuseofthetensornotation,andtheinterplaybetween
algebraandgeometry.Theearlychaptershavemanywordsandfewequations.The
definitionofatensorcomesonlyinChap.6—whenthereaderisreadyforit.
Part III of this book is devoted to the calculus of moving surfaces (CMS). One
of the central applications of tensor calculus is differential geometry, and there is
probably not one book about tensors in which a major portion is not devoted to
manifolds. The CMS extends tensor calculus to moving manifolds. Applications
of the CMS are extraordinarily broad. The CMS extends the language of tensors
to physical problems with moving interfaces. It is an effective tool for analyzing
boundaryvariationsofpartialdifferentialequations.Italsoenablesustobringthe
calculusofvariationswithinthetensorframework.
Whilethisbookmaintainsareasonablelevelofrigor,ittakesgreatcaretoavoid
a formalization of the subject. Topological spaces and groups are not mentioned.
Instead,thisbookfocusesonconcreteobjectsandappealstothereader’sgeometric
intuitionwithrespecttosuchfundamentalconceptsastheEuclideanspace,surface,
length,area,andvolume.Afewotherbooksdoagoodjobinthisregard,including
[2,8,31,46]. The book [42] is particularly concise and offers the shortest path to
the general relativity theory. Of course, for those interested in relativity, Hermann
v
vi Preface
Weyl’s classic Space, Time, Matter [47] is without a rival. For an excellent book
withanemphasisonelasticity,see[40].
Along with eschewing formalism, this book also strives to avoid vagueness
associatedwithsuchnotionsastheinfinitesimaldifferentialsdxi.Whileanumber
offundamentalconceptsareacceptedwithoutdefinition,allsubsequentelementsof
thecalculusarederivedinaconsistentandrigorousway.
The description of Euclidean spaces centers on the basis vectors Z . These
i
importantandgeometricallyintuitiveobjectsareabsentfrommanytextbooks.Yet,
their use greatly simplifies the introduction of a number of concepts, including
the metric tensor Z D Z (cid:2) Z and Christoffel symbol (cid:2)i D Zi (cid:2) @Z =@Zk.
ij i j jk j
Furthermore, the use of vector quantities goes a long way towards helping the
studentseetheworldinawaythatisindependentofCartesiancoordinates.
The notation is of paramount importance in mastering the subject. To borrow
a sentence from A.J. McConnell [31]: “The notation of the tensor calculus is so
muchanintegralpartofthecalculusthatoncethestudenthasbecomeaccustomed
to its peculiarities he will have gone a long way towards solving the difficulties
of the theory itself.” The introduction of the tensor technique is woven into the
presentationofthematerialinChap.4.Asaresult,theframeworkisdescribedina
naturalcontextthatmakestheeffectivenessoftherulesandconventionsapparent.
Thisisunlikemostothertextbookswhichintroducethetensornotationinadvance
oftheactualcontent.
In spirit and vision, this book is most similar to A.J. McConnell’s classic
ApplicationsofTensorCalculus[31].Hisconcreteno-frillsapproachisperfectfor
the subject and served as an inspiration for this book’s style. Tullio Levi-Civita’s
ownTheAbsoluteDifferentialCalculus[28]isanindispensablesourcethatreveals
themotivationsofthesubject’sco-founder.
Sinceaheavyemphasisinplacedonvector-valuedquantities,itisimportantto
havegoodfamiliaritywithgeometricvectorsviewedasobjectsontheirownterms
ratherthanelementsinRn.Anumberoftextbooksdiscussthegeometricnatureof
vectorsingreatdepth.FirstandforemostisJ.W.Gibbs’classic[14],whichservedas
aprototypeforlatertexts.Danielson[8]alsogivesagoodintroductiontogeometric
vectorsandoffersanexcellentdiscussiononthesubjectofdifferentiationofvector
fields.
Thefollowingbooksenjoyagoodreputationinthemoderndifferentialgeometry
community: [3,6,23,29,32,41]. Other popular textbooks, including [38,43] are
knownfortakingtheformalapproachtothesubject.
Virtually all books on the subject focus on applications, with differential
geometryfrontandcenter.Othercommonapplicationsincludeanalyticaldynamics,
continuummechanics,andrelativitytheory.Somebooksfocusonparticularappli-
cations. A case in point is L.V. Bewley’s Tensor Analysis of Electric Circuits And
Machines[1].Bewleyenvisionedthatthetensorapproachtoelectricalengineering
wouldbecomeastandard.Hereishopinghisdreameventuallycomestrue.
Philadelphia,PA PavelGrinfeld
Contents
1 WhyTensorCalculus? .................................................... 1
PartI TensorsinEuclideanSpaces
2 RulesoftheGame .......................................................... 11
2.1 Preview.............................................................. 11
2.2 TheEuclideanSpace ............................................... 11
2.3 Length,Area,andVolume.......................................... 13
2.4 ScalarsandVectors ................................................. 14
2.5 TheDotProduct .................................................... 14
2.5.1 InnerProductsandLengthsinLinearAlgebra......... 15
2.6 TheDirectionalDerivative ......................................... 15
2.7 TheGradient ........................................................ 17
2.8 DifferentiationofVectorFields ................................... 18
2.9 Summary ............................................................ 19
3 CoordinateSystemsandtheRoleofTensorCalculus.................. 21
3.1 Preview.............................................................. 21
3.2 WhyCoordinateSystems? ......................................... 21
3.3 WhatIsaCoordinateSystem? ..................................... 23
3.4 PerilsofCoordinates................................................ 24
3.5 TheRoleofTensorCalculus....................................... 27
3.6 ACatalogofCoordinateSystems.................................. 27
3.6.1 CartesianCoordinates ................................... 28
3.6.2 AffineCoordinates ...................................... 30
3.6.3 PolarCoordinates ....................................... 30
3.6.4 CylindricalCoordinates ................................. 31
3.6.5 SphericalCoordinates ................................... 32
3.6.6 RelationshipsAmongCommonCoordinate
Systems................................................... 32
3.7 Summary ............................................................ 34
vii
viii Contents
4 ChangeofCoordinates..................................................... 35
4.1 Preview.............................................................. 35
4.2 AnExampleofaCoordinateChange.............................. 35
4.3 AJacobianExample................................................ 36
4.4 TheInverseRelationshipBetweentheJacobians ................. 37
4.5 TheChainRuleinTensorNotation................................ 38
4.6 InverseFunctions ................................................... 42
4.7 InverseFunctionsofSeveralVariables ............................ 43
4.8 TheJacobianPropertyinTensorNotation......................... 45
4.9 SeveralNotesontheTensorNotation.............................. 48
4.9.1 TheNamingofIndices .................................. 48
4.9.2 CommutativityofContractions ......................... 49
4.9.3 MoreontheKroneckerSymbol......................... 50
4.10 Orientation-PreservingCoordinateChanges ...................... 50
4.11 Summary ............................................................ 51
5 TheTensorDescriptionofEuclideanSpaces............................ 53
5.1 Preview.............................................................. 53
5.2 ThePositionVectorR .............................................. 53
5.3 ThePositionVectorasaFunctionofCoordinates ................ 54
5.4 TheCovariantBasisZ ............................................. 55
i
5.5 TheCovariantMetricTensorZ .................................. 56
ij
5.6 TheContravariantMetricTensorZij .............................. 57
5.7 TheContravariantBasisZi ......................................... 58
5.8 TheMetricTensorandMeasuringLengths........................ 59
5.9 IntrinsicObjectsandRiemannSpaces............................. 60
5.10 DecompositionwithRespecttoaBasisbyDotProduct.......... 60
5.11 TheFundamentalElementsinVariousCoordinates .............. 62
5.11.1 CartesianCoordinates ................................... 62
5.11.2 AffineCoordinates....................................... 63
5.11.3 PolarandCylindricalCoordinates...................... 65
5.11.4 SphericalCoordinates ................................... 66
5.12 TheChristoffelSymbol(cid:2)k ........................................ 67
ij
5.13 TheOrderofIndices................................................ 71
5.14 TheChristoffelSymbolinVariousCoordinates................... 72
5.14.1 CartesianandAffineCoordinates....................... 72
5.14.2 CylindricalCoordinates ................................. 72
5.14.3 SphericalCoordinates ................................... 72
5.15 Summary ............................................................ 73
6 TheTensorProperty ....................................................... 75
6.1 Preview.............................................................. 75
6.2 Variants.............................................................. 75
6.3 DefinitionsandEssentialIdeas..................................... 76
6.3.1 TensorsofOrderOne.................................... 76
6.3.2 TensorsAretheKeytoInvariance...................... 76
Contents ix
6.3.3 TheTensorPropertyofZ ............................... 77
i
6.3.4 TheReverseTensorRelationship ....................... 78
6.3.5 TensorPropertyofVectorComponents................. 79
6.3.6 TheTensorPropertyofZi .............................. 80
6.4 TensorsofHigherOrder............................................ 80
6.4.1 TheTensorPropertyofZ andZij .................... 81
ij
6.4.2 TheTensorPropertyofıi ............................... 81
j
6.5 Exercises ............................................................ 82
6.6 TheFundamentalPropertiesofTensors ........................... 83
6.6.1 SumofTensors........................................... 83
6.6.2 ProductofTensors....................................... 83
6.6.3 TheContractionTheorem............................... 84
6.6.4 The Important Implications
oftheContractionTheorem............................. 85
6.7 Exercises ............................................................ 85
6.8 TheGradientRevisitedandFixed.................................. 86
6.9 TheDirectionalDerivativeIdentity................................ 87
6.10 IndexJuggling ...................................................... 88
6.11 TheEquivalenceofıi andZ ..................................... 90
j ij
6.12 TheEffectofIndexJugglingontheTensorNotation............. 91
6.13 Summary ............................................................ 91
7 ElementsofLinearAlgebrainTensorNotation ........................ 93
7.1 Preview.............................................................. 93
7.2 The Correspondence Between Contraction
andMatrixMultiplication .......................................... 93
7.3 TheFundamentalElementsofLinearAlgebra
inTensorNotation .................................................. 96
7.4 Self-AdjointTransformationsandSymmetry ..................... 99
7.5 QuadraticFormOptimization...................................... 101
7.6 TheEigenvalueProblem............................................ 103
7.7 Summary ............................................................ 104
8 CovariantDifferentiation ................................................. 105
8.1 Preview.............................................................. 105
8.2 AMotivatingExample.............................................. 106
8.3 TheLaplacian....................................................... 109
8.4 TheFormulaforr Z .............................................. 112
i j
8.5 TheCovariantDerivativeforGeneralTensors .................... 114
8.6 PropertiesoftheCovariantDerivative............................. 115
8.6.1 TheTensorProperty..................................... 115
8.6.2 TheCovariantDerivativeAppliedtoInvariants........ 117
8.6.3 TheCovariantDerivativeinAffineCoordinates ....... 117
8.6.4 Commutativity ........................................... 118
8.6.5 TheSumRule............................................ 119
8.6.6 TheProductRule ........................................ 119