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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

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

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.

( )

( )

( )

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

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 =

{ } { }

{ } { }

{ } { }

{ } { }

0-

1-

2-

3-

forms functions

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...