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  • MathematicalSupplementElectromagneticTheory(Physics704/804)

    SomeMathwellneedgoingforward

    Iwillassumeknowledgeofvectorsandvectorfields

    2tensors:A9componentobjectin3space(16componentobjectin4space)thattransformsassquaresofvectorsin3space(squaresof4vectorsin4space).

    Let v v v vx y zx y z= + +r

    in3space

    Formtheobject ( )( ) ( )( ) v v v v v v v v v v v v v vx y z x y z x x x y z zx y z x y z xx xy zz= + + + + + + +r r L

    andacorrespondencebetweenthesquaredobjectand3 3matrices(4 4matrices)as

    v v v v v v v v v v v v v v v v v v

    v v v v v v

    x x x y x z

    x x x y z z y x y y y z

    z x z y z z

    xx xy zz

    + + +

    L .

    Underachangeofbasis

    1 11 12 13

    2 21 22 23

    3 31 32 33

    11 1 12 1 33 3

    , , ,

    e E x E y E ze E x E y E ze E x E y E z

    E e x E e y E e z

    = + += + += + +

    = = =

    r

    r

    r

    r r rL

    withthematrix E nonsingular,torepresentthesameobject,thevectorcomponentsmustbetransformedby

    1

    12

    3

    v vv vv v

    xt

    y

    z

    E

    =

    ,

    wheretdenotesthetranspose,andthecomponentsofthesquaredquantitymustbetransformedas

    1 1v v v v v v v v v v v vv v v v v v v v v v v vv v v v v v v v v v v v

    x x x y x z x x x y x zt

    y x y y y z y x y y y z

    z x z y z z z x z y z z

    E E

    =

    ,

    becausethen

  • 1 1 1 1 1 2 1 2 3 3 3 3 v v v v v v v v v v v vx x x z z ze e e e e e xx xy zz + + + = + + +r r r r r r

    L L .

    Abstractingthisidea,a2tensorisany9component(16component)objectwhosecomponentstransformasthematrixequationaboveunderachangeofbasis.Thistypeofdefinitionoftensors,intermsoftransformationrules,isprevalentintheolderliterature,e.g.,Einsteinspapersongeneralrelativity.

    Goingforwardinthecourse,weshallgenerallywritetensorcomponentsintermsofthestandard

    ( ) , ,x y z basisor ( ) , , ,ct x y z 4basis.

    Thesquaringoperationaboveisanexampleofamoregeneralnotioncalledthetensorproduct.

    Asanexampleofthisproduct:if 1 1 1 1 v v v vx y zx y z= + +r

    and 2 2 2 2 v v v vx y zx y z= + +r

    arevectors,

    thenthequantity

    1 2 1 2 1 2 1 2 v v v v v v v vx x x y z zT xx xy zz= + + +r r

    K ,

    becausethebasisvectorsandvectorcomponentstransformexactlyasinsquaringabove,isaninecomponent2tensorin3space.Thetensorproduct, ,whichcanbegeneralizedbeyondthisspecificapplication,providesandverypowerfulwaytogeneratenew,higherranktensorsfromlowerranktensors.Thetensorproductislinearineachofitsarguments.

    Note,bydefinition,

    , , ,xx x x xy x y zz z z= = = K

    and x y y x .

    Notall2tensorscanbewritten 1 2v vr r

    forsome 1vrand 2v

    r(Exerciseforreader:whichonescanbeso

    written?),butallcanbewrittenas9sums(16sums)

  • 11 12 33 T t xx t xy t zz= + + +L

    forsomerealnumbers ijt ,calledthecomponentsofthetensorinthestandardbasis,where i

    and j extendfrom1to3(0to3).Toprovethisassertionnotethatifthe2tensorisdottedintothestandardbasisvectors

    11 12 33 , , , t x T x t x T y t z T z= = = L ,then 11 12 33 T t xx t xy t zz= + + +L .

    Thecomponentsaresometimeswritteninmatrixform

    11 12 13

    21 22 23

    31 32 33

    t t tt t tt t t

    ,orfor4space

    00 01 02 03

    10 11 12 13

    20 21 22 23

    30 31 32 33

    t t t tt t t tt t t tt t t t

    ,

    andmanipulatedasasingleentity.

    Relationshipbetween3by3matrices,2tensors,andbilinearmaps.Setupacorrespondence

    ( ) ( )( )

    ( ) ( )( )

    ( ) ( )

    1 2 1 2 1 2

    1 2 1 2 1 2

    1 2 1

    1 0 0 0 0 0 v , v v v v v

    0 0 0

    0 1 0 0 0 0 v , v v v v v

    0 0 0

    0 0 0 0 0 0 v , v v

    0 0 1

    xx x x

    xy x y

    zz

    xx T x x

    xy T x y

    zz T z z

    = = = =

    =

    r r r r

    r r r r

    M

    r r r ( )2 1 2v v vz z =r

    Eachofwhichisobviouslybilinear.Anybilinearmap 3 3: R R RL ,canberepresentedbyamatrix,andhencebyacorrespondingtensorbyexpandinganalogouslytoabove.Let

    ( ) ( ) ( )11 12 33 , , , , , ,l L x x l L x y l L z z= = =L ,then 11 12 33 L l xx l xy l zz= + + +L

  • isthecorresponding2tensorthatgivesthemapby ( )1 2 1 2v , v v vL L= r r r r

    (verifythisassertionfor

    arbitrary 1vrand 2v

    r).

    Ageneral2tensordefinesabilinearmap ( )3 3 1 2 1 2: R R R, v , v v vT T T = r r r r

    ,which,bythe

    tensorcomponenttransformationrule,doesnotdependonthebasisinwhichitisevaluated(verify!).Forthestandardbasisitevaluatesto

    ( )1 2 11 1 2 12 1 2 33 1 2v , v v v v v v vx x x y z zT t t t= + + +r r

    L .

    Note

    ( ) ( ) ( ) ( )

    ( ) ( )

    1 2 11 1 2 12 1 2 33 1 2

    11 1 11 2 12 1 12 2 33 1 33 2

    1 2

    v v , v v v v v v v v v v

    v v v v v v v v v v v v

    v , v v , v ,

    x x x x x y z z z

    x x x x x y x y z z z z

    T t t t

    t t t t t t

    T T

    + = + + + + + +

    = + + + + + +

    = +

    r r rL

    Lr r r r

    andsimilarly ( ) ( ) ( )1 2 1 2v, v v v, v v, vT T T + = +r r r r r r r

    .Sothereisadirectonetoonecorrespondence

    between2tensorsandbilinearmaps.Someauthorsusethisequivalencetodefinetensorsverygenerallyasmultilinearmapsonparticularvectorspaces.Wewillnotneedtoemploythisfullgenerally.

    Atensorissymmetricorantisymmetricdependingonwhethertherepresentingmatrixissymmetricorantisymmetric.Itisstraightforwardtoshow,usingthechangeofbasisformula,thatthisisaframeinvariantnotion.

    Modernnotation:Letthecoordinatesof 3R begivenby 1 2 3, ,x x x y x z= = = .Followingstandard

    sloppybehaviornotdistinguishingthespace 3R fromitstangentspace,definethexcomponentoperatorby

    ( ) ( )1 v v v vxdx dx x= = =r r r

    .

    Thisisalinearmapinthetangentspaceto 3R ,whichfollowingstandardprocedureisidentifiedwith3R ,anddefinedregardlessofwherein 3R thevectorissituated.Likewisedefine

    ( ) ( )( ) ( )

    2

    3

    v v v v

    v v v v .y

    z

    dx dy y

    dx dz z

    = = =

    = = =

    r r r

    r r r

    Withthenumberingconventionabove,itisclear(tracethisthrough!)thatthegeneral3space(4space)2tensorisgivenas

  • ,i jijT t dx dx=

    where i and j aresummedfrom1to3(0to3).Goingforward,thefollowingEinsteinsummationwillbefollowed.Whenanupperandlowerlatinindexisused,thesummationindexgoesfrom1to3.Whenanupperandlowergreekindexisused,thesumproceedsfrom0(representingthetimecoordinate)to3.

    In3space,theantisymmetricaltensorshaveanadditionalspecializednotation

    ( )

    ( )

    ( )

    2 3 2 3 3 21 2 1 2 1 2

    3 1 3 1 1 31 2 1 2 1 2

    1 2 1 2 2 11 2

    0 0 00 0 1 v , v v v v v0 1 0

    0 0 10 0 0 v , v v v v v1 0 0

    0 1 01 0 0 v , v

    0 0 0

    y z z y

    z x x z

    dx dx dx dx dx dx dy dz

    dx dx dx dx dx dx dz dx

    dx dx dx dx dx dx dx dy

    = =

    = = = =

    r r

    r r

    r r1 2 1 2v v v vx y y x

    Theseformulasshouldremindoneoftheprojectionofthevectorcrossproduct(withpropersign)ontothemissingcoordinateaxes.Also,onecangiveamorephysicalinterpretationoftheantisymmetric

    product. dx dy operatingonthepair ( )1 2v , vr r

    givestheareaoftheparallelepipedwhoseedgesare

    formedbytheprojectionofthetwovectorsintothexyplane.Likewise dy dz givestheareaoftheparallelepipedwhoseedgesareformedbytheprojectionofthetwovectorsintotheyzplane,and

    dz dx theareaafterprojectionintothezxplane.Theorderisimportant,positivemeanstheprojectionof 1v

    r,theprojectionof 2v

    r,andtheremainingpositiveaxisdirectionformarighthandedset

    ofvectors.Alsoclearly,anyantisymmetrictensorcanbewrittenasthesumofthesethreebasistensors.Forfourspace,2tensorshave16componentsandtheantisymmetictensorshaveuptosixindependentcomponents.Inthiscase,thebasicantisymmetrictensorsare

  • 1 1 1

    2 2 2

    0 1 0 01 0 0 0

    0 0 0 00 0 0 0

    0 0 1 00 0 0 01 0 0 0

    0 0 0 0

    cdt dx cdt dx cdx dt

    cdt dx cdt dx cdx dt

    = =

    3 3 3

    1 2 1 2 2 1

    2 3 2 3 3 2

    3 1 3 1 1 3

    0 0 0 10 0 0 00 0 0 01 0 0 0

    0 0 0 00 0 1 00 1 0 00 0 0 0

    0 0 0 00 0 0 00 0 0 10 0 1 0

    0 0 0 00 0 0 10 0 0 00 1 0 0

    cdt dx cdt dx cdx dt

    dx dx dx dx dx dx

    dx dx dx dx dx dx

    dx dx dx dx dx dx

    =

    = =

    =

    DifferentialForms

    Inthiscoursewellrestrictourselvestodifferentialformsonflat3space,oronspecialrelativitysMinkowskispace.Thesamegeneralconstructionscanbeused,withevenmorepower,ongeneralmanifoldsandthecurvedspacetimesofgeneralrelativity.Thesecomplicationsarenotusefulinacourseonelectrodynamics.Also,wewillgenerallyassumethatfunctionshaveasmanyderivativesasnecessarytoensurederivationsarevalid.

  • Adifferentiablefunction 3: R Rf willbecalleda0form.

    A1formisanymapping,linearonvectorspaces,oftheform

    ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( )( )

    1 1 21 2

    33

    v , , , , v , , , , v , ,

    , , v , , ,

    x y z a x y z dx x y z a x y z dx x y z

    a x y z dx x y z

    = +

    +

    r r r

    r ,

    wherethe ia aredifferentiablefunctions,andasabove,theidx projectoutspecificcomponentsofthe

    vectorfieldatthelocationinquestion.1formsoperateonvectorfieldsandyieldascalar(i.e.,basisindependent)0formonevaluationwithaspecificvectorfield.

    A2formistheanalogousconstructionappliedtoantisymmetric2tensorfields.Itisabilinearantisymmetricmappingoftheform

    ( ) ( ) ( )( )

    2 2 3 3 11 2 1 1 2 2 1 2

    1 23 1 2

    v , v v , v v , v

    v , v ,

    a dx dx a dx dx

    a dx dx

    = +

    +

    r r r r r r

    r r

    where,forconvenienceofnotation,the ( ), ,x y z dependenceofthe ia andthetwovectorfieldsissuppressed.Itwasnotedabovethatanyantisymmetric2tensorfieldmustbeofthisform.

    A3form,ortotallyantisymmetic3tensorfield,isgivenas

    ( ) ( )

    ( )

    3 1 2 31 2 3 1 2 3

    1 2 3 2 3 1 3 1 2

    1 2 31 3 2 2 1 3 3 2 1

    v , v , v v , v , v

    v , v , v

    adx dx dx

    dx dx dx dx dx dx dx dx dxa

    dx dx dx dx dx dx dx dx dx

    = =

    + +

    r r r r r r

    r r r

    wherea isadifferentiablefunction.Whenevaluatedonthetriple ( )1 2 3v , v , vr r r

    , 3 / a yieldstheso

    calledtriplescalarproductofthevectors 1 2 3v v v r r r

    ,whichistheorientedvolumeoftheparallelliped

    withsidesgivenbythethreevectors.Therearenohigherformsin3space.Itisanexercisetowriteoutthecorrespondingforms,includingthepossibilityof4forms,inMinkowskispace.

    ExteriorDerivativeMap d

    Inthisdocument,theexteriorderivativewillbedefinedoperationally.Refertoanygoodbookondifferentialformsformorecompletederivations.Theexteriorproductalwayscarriesan i formtoan

    1i + formandisevaluatedusingseveralbasicrules

    1. f f fdf dx dy dzx y z

    = + +

    whenevaluatedonazeroform.Thelinearmap ( )vdf r isbetter

    knownasthedirectionalderivativeofthefunction f inthedirection vr .Notethatwhen

  • evaluatedonthecoordinatefunctions,itevaluatestothecorrectvalue.Forexample1 0 0dx dx dy dz= + + .

    2. ( ) ( )1 id a b da b a db = + wherea isan i form,themodifiedLeibnitzrule3. Theantisymmetricproduct isanticommutativeon1forms(showthis!!)andthus

    0dx dx dy dy dz dz = = =

    4. ( )2 0d d d = foranyformbecausemixedpartialderivativesarealwaysequal

    Examples:Supposethegeneral1formiswritteninthefollowingsuggestiveway

    1 1 2 31 2 3 .a dx a dx a dx = + +

    Then

    1 1 2 3 11 1 11 2 3

    1 2 3 22 2 21 2 3

    1 2 3 33 3 31 2 3

    3

    0

    0

    0

    a a ad dx dx dx dxx x xa a adx dx dx dxx x xa a adx dx dx dxx x x

    ax

    = + + + + + + + + + + + =

    2 3 3 1 1 232 1 2 12 3 3 1 1 2 .

    aa a a adx dx dx dx dx dxx x x x x

    + +

    Likewiseif

    2 2 3 3 1 1 21 2 3 ,a dx dx a dx dx a dx dx = + +

    2 1 2 3 2 31 1 11 2 3

    1 2 3 3 12 2 21 2 3

    1 2 3 1 23 3 31 2 3

    31 21 2 3

    0

    0

    0

    a a ad dx dx dx dx dxx x xa a adx dx dx dx dxx x xa a adx dx dx dx dxx x x

    aa ax x x

    = + + + + + + + + + + + =

    + + 1 2 3dx dx dx

    Ifaregularvectorisassociatedwitha1form

    1v vi

    idx = ,

    d generatesthe(polarvector!)curlofthevectorfield.Ifapolarvectorisassociatedwitha2form

  • 2v v ,j k

    ijk idx dx =

    d generatesthedivergenceofthestartingvectorfield.Therefore,thestandardgradient,divergence,andcurloperationsinvectoranalysisareincludedascasesofthe d operationsinforms.Wewilltaketheapproachthattheusualvectoranalysisismostnaturallyanalyzedwithmathematicsofforms.Thefollowingcorrespondencesapply

    { } { }

    { } { }

    { } { }

    { } { }

    0-

    1-

    2-

    3-

    forms functions

    d gradforms vector fields

    d curlforms polar vector fields

    d divforms functions

    Thefactthat 2 0d = isequivalenttothevectoranalysisstatementsthat ( ) 0f = and( )v 0 =r .

    Formssuchthat 0d = arecalledclosed.Forms thatcanbewrittenastheexteriorderivativeofanother,lowerdegree,form da = ,arecalledexact.Itisamathematicalresult,calledthePoincarLemma,thatbecauseR3issimplyconnected(contractibleisenough),allclosedformsareexactin3space.ThesameresultappliesinMinkowskispace.InE&Mtheorythisfactisusedconstantly,particularlywhenthescalarpotentialandvectorpotentialfunctionsareintroducedtorepresentelectromagneticfields.

    IntegrationandgeneralizedStokesTheorem.

    0formscannotbeintegrated,onlyevaluatedatspecificlocations.

    1formsareintegratedaslineintegrals.LetC beacurve,openorclosedparameterizedbyanindependentvariable t .Then

    ( )( ) ( )( ) ( )( )1 1 2 3 .C C

    dx dy dza x t a x t a x t dtdt dt dt

    + +

    r r r

    Forexactforms 1 d = ,theintegralevaluatesasthedifferenceofthefunction

    value ( )( ) ( )( )x t x t r r attheendpointsofthecurve, ( )x tr and ( )x tr .Thisisthecaseofaconservativeforcefieldwith thepotentialfunction.Foraclosedcurvethelineintegralevaluatestozeroforexact1forms.

  • 2formsareintegratedassurfaceintegrals.Let S beasurfacewritten,forexample,as

    ( )( ), , ,x y z s x y= .Now s sdz dx dyx y

    = +

    ,so

    ( )( ) ( )( ) ( )( )

    ( )( ) ( )( ) ( )( )

    21 2 3

    1 2 3( )

    , , , , , , , , ,

    , , , , , , , , ,

    S S

    p S

    s sa x y s x y dy dx a x y s x y dy dx a x y s x y dx dyx y

    s sa x y s x y a x y s x y a x y s x y dxdyx y

    + +

    = +

    where ( )p S istheprojectedareaofthesurfaceintothe x y planeandthefinalintegralistheusual2Dmultipleintegral.Notethatfortheintegraltomakesense, s mustbesinglevaluedwithintheintegrationdomain ( )p S .Totalsurfaceintegralsmustbebrokenupintosumsofsinglevaluedportionsincases,suchaswhenthesurfaceisclosed,where z isnotuniquelydeterminedby x and y .Italsomaybeconvenient,andispossible,toexpresssomeportionsofthesurfaceusingtheindependentvariables x and z or y and z ,orevenusingparametricrepresentationsofthesurface,inordertoevaluatepartsof,orthecompletesurfaceintegral.

    Now ( )/ , / ,1n z x z y= r isinthedirectionofthenormaltothesurface.So

    ( ) ( )2( )

    S p S S

    a n n dxdy a n dA = = r r r

    where dA isthelocalareaelementofthesurface,thatisprojecteddownintotheareaelementdxdy bythemagnitudeof nr .Thisfinalformoftheintegralisthatusuallyfoundinvectoranalysis.

    Fluxintegralsareperformedbydoingsuchsurfaceintegrals.Thetotalsurfaceintegralevaluatestozeroonaclosedsurface(i.e.,asurfacewithnoboundary)ifthe2formisexact.

    3formsareintegratedassignedvolumeintegrals

    ( ) ( )3 , , , , .V V V

    a x y z dx dy dz a x y z dxdydz =

    wherethefinalintegralistheusual3dmultipleintegral.

    GeneralizedStokesTheorem(foraproof,andconditionsforvalidity,seeanygoodbookonforms)

  • IfM isamanifoldand aformthatexiststhroughoutthemanifold,then

    M M

    d

    =

    where M istheboundaryofM .Wewillbeapplyingthistheoremtovariousregionsof3spaceandMinkowskispace,althoughitworksfor,andinsideofmuchmoregeneralmanifolds.Withinitarethemaintheoremsofvectoranalysis.

    FundamentalTheoremofCalculus:Appliedto[a,b]inR .If f =

    [ ]( ) ( )

    ,

    b

    aa b

    df dx f f b f adx

    = =

    GreensTheorem:AppliedtoasimplyconnectedregionRof 2R .If Pdx Qdy = +

    ( )( )

    / .R R p R

    Q P dxdy Pdx Qdy P Qdy dx dxx y

    = + = +

    Thesecondequalityappliesinthecasethat ( )y x isthemonotonic,andsinglevaluedequationfortheboundarycurveintermsoftheindependentvariable x .Notethatif 0d = ,i.e.the...

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