張量分析翻譯 英文原文

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張量分析翻譯英文原文

Tensor

Tensorsaregeometricobjectsthatdescribelinear

relationsbetweenvectors,scalars,andothertensors.

Elementaryexamplesofsuchrelationsincludethe

dotproduct,thecrossproduct,andlinear

maps.Vectorsandscalarsthemselvesarealsotensors.

Atensorcanberepresentedasamulti-dimensional

arrayofnumericalvalues.Theorder(alsodegreeor

rank)ofatensoristhedimensionalityofthearray

neededtorepresentit,orequivalently,thenumberof

indicesneededtolabelacomponentofthatarray.Forexample,alinearmapcanberepresentedbyamatrix,a2-dimensionalarray,andthereforeisa2nd-ordertensor.Avectorcanberepresentedasa1-dimensionalarrayandisa

1st-ordertensor.Scalarsaresinglenumbersand

arethus0th-ordertensors.

Tensorsareusedtorepresentcorrespondencesbetweensetsofgeometricvectors.Forexample,theCauchystresstensorTtakesadirectionvasinputandproducesthestressT(v)onthesurface

normaltothisvectorforoutputthusexpressing

arelationshipbetweenthesetwovectors,showninthefigure(right).

Becausetheyexpressarelationshipbetweenvectors,tensorsthemselvesmustbe

independentofaparticularchoiceofcoordinatesystem.Takingacoordinatebasisorframeofreferenceandapplyingthetensortoitresultsinanorganizedmultidimensionalarrayrepresentingthetensorinthatbasis,orframeofreference.Thecoordinateindependenceofatensorthentakestheformofa"covariant"transformationlawthatrelatesthearraycomputedinonecoordinatesystemtothatcomputedinanotherone.Thistransformationlawisconsideredtobebuiltintothenotionofatensorinageometricorphysicalsetting,andthepreciseformofthetransformationlawdeterminesthetype(orvalence)ofthetensor.

Tensorsareimportantinphysicsbecausetheyprovideaconcisemathematicalframeworkforformulatingandsolvingphysicsproblemsinareassuchaselasticity,fluidmechanics,andgeneralrelativity.TensorswerefirstconceivedbyTullioLevi-CivitaandGregorioRicci-Curbastro,whocontinuedtheearlierworkofBernhardRiemannandElwinBrunoChristoffelandothers,aspartoftheabsolutedifferentialcalculus.TheconceptenabledanalternativeformulationoftheintrinsicdifferentialgeometryofamanifoldintheformoftheRiemanncurvaturetensor.[1]Cauchystresstensor,asecond-ordertensor.Thetensor'scomponents,inathree-dimensionalCartesiancoordinatesystem,formthematrixwhosecolumnsarethestresses(forcesperunitarea)actingonthee1,e2,ande3facesofthecube.

History

TheconceptsoflatertensoranalysisarosefromtheworkofCarlFriedrichGaussindifferentialgeometry,andtheformulationwasmuchinfluencedbythetheoryofalgebraicformsandinvariantsdevelopedduringthemiddleofthenineteenthcentury.[2]Theword"tensor"itselfwasintroducedin1846byWilliamRowanHamilton[3]todescribesomethingdifferentfromwhatisnowmeantbyatensor.[Note1]ThecontemporaryusagewasbroughtinbyWoldemarVoigtin1898.[4]

Tensorcalculuswasdevelopedaround1890byGregorioRicci-Curbastrounderthetitleabsolutedifferentialcalculus,andoriginallypresentedbyRicciin1892.[5]ItwasmadeaccessibletomanymathematiciansbythepublicationofRicciandTullioLevi-Civita's1900classictextMéthodesdecalculdifférentielabsoluetleursapplications(Methodsofabsolutedifferentialcalculusandtheirapplications).[6]

Inthe20thcentury,thesubjectcametobeknownastensoranalysis,andachievedbroaderacceptancewiththeintroductionofEinstein'stheoryofgeneralrelativity,around1915.Generalrelativityisformulatedcompletelyinthelanguageoftensors.Einsteinhadlearnedaboutthem,withgreatdifficulty,fromthegeometerMarcelGrossmann.[7]Levi-CivitatheninitiatedacorrespondencewithEinsteintocorrectmistakesEinsteinhadmadeinhisuseoftensoranalysis.Thecorrespondencelasted1915–17,andwascharacterizedbymutualrespect:

Iadmiretheeleganceofyourmethodofcomputation;itmustbenicetoridethroughthesefieldsuponthehorseoftruemathematicswhilethelikeofushavetomakeourwaylaboriouslyonfoot.

—AlbertEinstein,TheItalianMathematiciansofRelativity[8]

Tensorswerealsofoundtobeusefulinotherfieldssuchascontinuummechanics.Somewell-knownexamplesoftensorsindifferentialgeometryarequadraticformssuchasmetrictensors,andtheRiemanncurvaturetensor.TheexterioralgebraofHermannGrassmann,fromthemiddleofthenineteenthcentury,isitselfatensortheory,andhighlygeometric,butitwassometimebeforeitwasseen,withthetheoryofdifferentialforms,asnaturallyunifiedwithtensorcalculus.TheworkofélieCartanmadedifferentialformsoneofthebasickindsoftensorsusedinmathematics.Fromaboutthe1920sonwards,itwasrealisedthattensorsplayabasicroleinalgebraictopology(forexampleintheKünneththeorem).[citationneeded]Correspondinglytherearetypesoftensorsatworkinmanybranchesofabstractalgebra,particularlyinhomologicalalgebraandrepresentationtheory.Multilinearalgebracanbedevelopedingreatergeneralitythanforscalarscomingfromafield,butthetheoryisthencertainlylessgeometric,andcomputationsmoretechnicalandlessalgorithmic.[clarificationneeded]Tensorsaregeneralizedwithincategorytheoryby

meansoftheconceptofmonoidalcategory,fromthe1960s.

Definition

Thereareseveralapproachestodefiningtensors.Althoughseeminglydifferent,theapproachesjustdescribethesamegeometricconceptusingdifferentlanguagesandatdifferentlevelsofabstraction.

Asmultidimensionalarrays

Justasascalarisdescribedbyasinglenumber,andavectorwithrespecttoagivenbasisisdescribedbyanarrayofonedimension,anytensorwithrespecttoabasisisdescribedbyamultidimensionalarray.Thenumbersinthearrayareknownasthescalarcomponentsofthetensororsimplyitscomponents.Theyaredenotedbyindicesgivingtheirpositioninthearray,insubscriptandsuperscript,afterthesymbolicnameofthetensor.Thetotalnumberofindicesrequiredtouniquelyselecteachcomponentisequaltothedimensionofthearray,andiscalledtheorderortherankofthetensor.[Note2]Forexample,theentriesofanorder2tensorTwouldbedenotedTij,whereiandjareindicesrunningfrom1tothedimensionoftherelatedvectorspace.[Note3]

Justasthecomponentsofavectorchangewhenwechangethebasisofthevectorspace,theentriesofatensoralsochangeundersuchatransformation.Eachtensorcomesequippedwithatransformationlawthatdetailshowthecomponentsofthetensorrespondtoachangeofbasis.Thecomponentsofavectorcanrespondintwodistinctwaystoachangeofbasis(seecovarianceandcontravarianceofvectors),

wherethenewbasisvectorsareexpressedintermsoftheoldbasisvectorsas,

whereRijisamatrixandinthesecondexpressionthesummationsignwassuppressed(anotationalconvenienceintroducedbyEinsteinthatwillbeusedthroughoutthisarticle).Thecomponents,vi,ofaregular(orcolumn)vector,v,transformwiththeinverseofthematrixR,

wherethehatdenotesthecomponentsinthenewbasis.Whilethecomponents,wi,ofacovector(orrowvector),wtransformwiththematrixRitself,

Thecomponentsofatensortransforminasimilarmannerwithatransformationmatrixforeachindex.Ifanindextransformslikeavectorwiththeinverseofthebasistransformation,itiscalledcontravariantandistraditionallydenotedwithanupperindex,whileanindexthattransformswiththebasistransformationitselfiscalledcovariantandisdenotedwithalowerindex.Thetransformationlawforanorder-mtensorwithncontravariantindicesandm?ncovariantindicesisthusgivenas,

Suchatensorissaidtobeoforderortype(n,m?n).[Note4]Thisdiscussionmotivatesthefollowingformaldefinition:[9]

Definition.Atensoroftype(n,m?n)isanassignmentofamultidimensionalarray

toeachbasisf=(e1,...,eN)suchthat,ifweapplythechangeofbasis

thenthemultidimensionalarrayobeysthetransformationlaw

ThedefinitionofatensorasamultidimensionalarraysatisfyingatransformationlawtracesbacktotheworkofRicci.[1]Nowadays,thisdefinitionisstillusedinsomephysicsandengineeringtextbooks.[10][11]

Tensorfields

Mainarticle:Tensorfield

Inmanyapplications,especiallyindifferentialgeometryandphysics,itisnaturaltoconsideratensorwithcomponentswhicharefunctions.Thiswas,infact,thesettingofRicci'soriginalwork.Inmodernmathematicalterminologysuchanobjectiscalledatensorfield,buttheyareoftensimplyreferredtoastensorsthemselves.[1]

Inthiscontextthedefiningtransformationlawtakesadifferentform.The"basis"forthetensorfieldisdeterminedbythecoordinatesoftheunderlyingspace,andthe

definingtransformationlawisexpressedintermsofpartialderivativesofthe

coordinatefunctions,,definingacoordinatetransformation,[1]

Asmultilinearmaps

Adownsidetothedefinitionofatensorusingthemultidimensionalarrayapproachisthatitisnotapparentfromthedefinitionthatthedefinedobjectisindeedbasisindependent,asisexpectedfromanintrinsicallygeometricobject.Althoughitispossibletoshowthattransformationlawsindeedensureindependencefromthebasis,sometimesamoreintrinsicdefinitionispreferred.Oneapproachistodefineatensorasamultilinearmap.Inthatapproachatype(n,m)tensorTisdefinedasamap,

whereVisavectorspaceandV*isthecorrespondingdualspaceofcovectors,whichislinearineachofitsarguments.

ByapplyingamultilinearmapToftype(n,m)toabasis{ej}forVandacanonicalcobasis{εi}forV*,

ann+mdimensionalarrayofcomponentscanbeobtained.Adifferentchoiceofbasiswillyielddifferentcomponents.But,becauseTislinearinallofitsarguments,thecomponentssatisfythetensortransformationlawusedinthemultilineararraydefinition.ThemultidimensionalarrayofcomponentsofTthusformatensoraccordingtothatdefinition.Moreover,suchanarraycanberealisedasthecomponentsofsomemultilinearmapT.Thismotivatesviewingmultilinearmapsastheintrinsicobjectsunderlyingtensors.

Usingtensorproducts

Mainarticle:Tensor(intrinsicdefinition)

Forsomemathematicalapplications,amoreabstractapproachissometimesuseful.Thiscanbeachievedbydefiningtensorsintermsofelementsoftensorproductsofvectorspaces,whichinturnaredefinedthroughauniversalproperty.Atype(n,m)tensorisdefinedinthiscontextasanelementofthetensorproductofvector

spaces,[12]

IfviisabasisofVandwjisabasisofW,thenthetensorproducthasa

naturalbasis.ThecomponentsofatensorTarethecoefficientsofthetensorwithrespecttothebasisobtainedfromabasis{ei}forVanditsdual{εj},i.e.

Usingthepropertiesofthetensorproduct,itcanbeshownthatthesecomponentssatisfythetransformationlawforatype(m,n)tensor.Moreover,theuniversalpropertyofthetensorproductgivesa1-to-1correspondencebetweentensorsdefinedinthiswayandtensorsdefinedasmultilinearmaps.

Operations

Thereareanumberofbasicoperationsthatmaybeconductedontensorsthatagainproduceatensor.Thelinearnatureoftensorimpliesthattwotensorsofthesametypemaybeaddedtogether,andthattensorsmaybemultipliedbyascalarwithresultsanalogoustothescalingofavector.Oncomponents,theseoperationsaresimplyperformedcomponentforcomponent.Theseoperationsdonotchangethetypeofthetensor,howevertherealsoexistoperationsthatchangethetypeofthetensors.

Raisingorloweringanindex

Mainarticle:Raisingandloweringindices

Whenavectorspaceisequippedwithaninnerproduct(ormetricasitisoftencalledinthiscontext),operationscanbedefinedthatconvertacontravariant(upper)indexintoacovariant(lower)indexandviceversa.Ametricitselfisa(symmetric)(0,2)-tensor,itisthuspossibletocontractanupperindexofatensorwithoneoflowerindicesofthemetric.Thisproducesanewtensorwiththesameindexstructureastheprevious,butwithlowerindexinthepositionofthecontractedupperindex.Thisoperationisquitegraphicallyknownasloweringanindex.

Converselythematrixinverseofthemetriccanbedefined,whichbehavesasa(2,0)-tensor.Thisinversemetriccanbecontractedwithalowerindextoproduceanupperindex.Thisoperationiscalledraisinganindex.

Applications

Continuummechanics

Importantexamplesareprovidedbycontinuummechanics.Thestressesinsideasolidbodyorfluidaredescribedbyatensor.Thestresstensorandstraintensorarebothsecondordertensors,andarerelatedinagenerallinearelasticmaterialbyafourth-orderelasticitytensor.Indetail,thetensorquantifyingstressina3-dimensionalsolidobjecthascomponentsthatcanbeconvenientlyrepresentedasa3×3array.Thethreefacesofacube-shapedinfinitesimalvolumesegmentofthesolidareeachsubjecttosomegivenforce.Theforce'svectorcomponentsarealsothreeinnumber.Thus,3×3,or9componentsarerequiredtodescribethestressatthiscube-shapedinfinitesimalsegment.Withintheboundsofthissolidisawholemassofvaryingstressquantities,eachrequiring9quantitiestodescribe.Thus,asecondordertensorisneeded.

Ifaparticularsurfaceelementinsidethematerialissingledout,thematerialononesideofthesurfacewillapplyaforceontheotherside.Ingeneral,thisforcewillnotbeorthogonaltothesurface,butitwilldependontheorientationofthesurfaceinalinearmanner.Thisisdescribedbyatensoroftype(2,0),inlinearelasticity,ormorepreciselybyatensorfieldoftype(2,0),sincethestressesmayvaryfrompointtopoint.

Otherexamplesfromphysics

Commonapplicationsinclude

?Electromagnetictensor(orFaraday'stensor)inelectromagnetism

?Finitedeformationtensorsfordescribingdeformationsandstraintensorforstrainincontinuummechanics

?Permittivityandelectricsusceptibilityaretensorsinanisotropicmedia

?Four-tensorsingeneralrelativity(e.g.stress-energytensor),usedtorepresentmomentumfluxes

?Sphericaltensoroperatorsaretheeigenfunctionsofthequantumangularmomentumoperatorinsphericalcoordinates

?Diffusiontensors,thebasisofDiffusionTensorImaging,representratesofdiffusioninbiologicenvironments

?QuantumMechanicsandQuantumComputingutilisetensorproductsforcombinationofquantumstates

Applicationsoftensorsoforder>2

Theconceptofatensorofordertwoisoftenconflatedwiththatofamatrix.Tensorsofhigherorderdohowevercaptureideasimportantinscienceandengineering,ashasbeenshownsuccessivelyinnumerousareasastheydevelop.Thishappens,forinstance,inthefieldofcomputervision,withthetrifocaltensorgeneralizingthefundamentalmatrix.

Thefieldofnonlinearopticsstudiesthechangestomaterialpolarizationdensityunder

extremeelectricfields.Thepolarizationwavesgeneratedarerelatedtothegeneratingelectricfieldsthroughthenonlinearsusceptibilitytensor.IfthepolarizationPisnotlinearlyproportionaltotheelectricfieldE,themediumistermednonlinear.Toagoodapproximation(forsufficientlyweakfields,assumingnopermanentdipolemomentsarepresent),PisgivenbyaTaylorseriesinEwhosecoefficientsarethenonlinearsusceptibilities:

Hereisthelinearsusceptibility,givesthePockelseffectandsecond

harmonicgeneration,andgivestheKerreffect.Thisexpansionshowsthewayhigher-ordertensorsarisenaturallyinthesubjectmatter.

Generalizations[edit]

Tensorsininfinitedimensions

Thenotionofatensorcanbegeneralizedinavarietyofwaystoinfinitedimensions.One,forinstance,isviathetensorproductofHilbertspaces.[15]Anotherwayofgeneralizingtheideaoftensor,commoninnonlinearanalysis,isviathemultilinearmapsdefinitionwhereinsteadofusingfinite-dimensionalvectorspacesandtheiralgebraicduals,oneusesinfinite-dimensionalBanachspacesandtheircontinuousdual.[16]TensorsthuslivenaturallyonBanachmanifolds.[17]

Tensordensities

Mainarticle:Tensordensity

Itisalsopossibleforatensorfieldtohavea"density".Atensorwithdensityrtransformsasanordinarytensorundercoordinatetransformations,exceptthatitisalsomultipliedbythedeterminantoftheJacobiantotherthpower.[18]Invariantly,inthelanguageofmultilinearalgebra,onecanthinkoftensordensitiesasmultilinearmapstakingtheirvaluesinadensitybundlesuchasthe(1-dimensional)spaceofn-forms(wherenisthedimensionoft