vv v v v v v v vv vv vv () ( ) - Center for Advanced Studies of casa.jlab.org/publications/viewgraphs/lecture/EMag Math ,v v v v v v v v v v vv v v vv v v vv v v v ,v v ,v , xxx x xy z zz xx xx x y x y z z z z Tt t t tt t t t t TT λλ λ

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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

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