4 Differential Forms and Electromagnetic Materials.pdf

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1、FilippoCapolino/TheoryandPhenomenaofMetamaterials_CFinalsPage--#4DifferentialFormsandElectromagneticMaterials.Introduction.........................................-.FieldandMediumEquations.........................-.ClassesofElectromagneticMed

2、ia....................-PerfectElectromagneticConductor●Q-Media●GeneralizedQ-Media●IB-Media●Self-DualMedia.Conclusion...........................................-Appendix:Multivectors,Multiforms,andDyadics.........-Notation●Products●Dyadics●ProductsofDyadics●Is

3、moV.LindellIdentitiesHelsinkiUniversityofTechnologyReferences.................................................-4.1IntroductionDifferential-formcalculusisabranchofmathematicsbasedonthealgebraofmultivectors(ele-mentsofspacesE,...,En)andmultiforms(dualmultivectors,ele

4、mentsofspacesF,...,Fn)[].ThenotationappliedherefollowscloselythatofRef.[]andashortsummaryisgivenintheAppendix.Correspondingrepresentationusingtensorsinsteadofdyadicscanbefound,e.g.,inRef.[].ApplicationofdifferentialformstoelectromagnetictheoryinsteadoftheclassicalG

5、ibbsianvectorformalism[,]issuggestedbythesimplicityandeleganceobtainedinwritingthebasicMaxwellequationsas[]d∧Φ=γm,d∧Ψ=γe.(.)Hered=∑ei∂xi=ds+e∂x(.)i=isthefour-dimensional(D)differentialoperatorand∧theexteriorproduct,whileΦ=B+E∧ε,Ψ=D−H∧ε(.)representtheDe

6、lectromagnetictwo-forms(fieldsdependingonthespatialcoordinatesx,x,xandthetemporalcoordinatex=τ=ct)intermsofspatial(D)two-formsB,Dandone-formsE,H.Theelectricandmagneticsourcethree-formsγe,γmcanbeexpressedasγe=ρe−Je∧ε,γm=ρm−Jm∧ε(.)intermsofcombinationsofspatial(

7、D)chargethree-formsρe,ρmandcurrenttwo-formsJe,Jm.4-1©2009byTaylorandFrancisGroup,LLCFilippoCapolino/TheoryandPhenomenaofMetamaterials_CFinalsPage--#4-2TheoryandPhenomenaofMetamaterialsInsertingtheDexpansions(Equations.and.),theDMaxwellequations(

8、Equation.)canbesplitinmorefamiliar-lookingDequationsasds∧E+∂τB=−Jm,ds∧B=ρm,(.)ds∧H−∂τD=Je,ds∧D=ρe.(.)Heredsden