Electromagnetic Green Functions Using Diff Forms-95--Warnick-p17--Pirx.pdf
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1、ElectromagneticGreenFunctionsUsingDi?erentialForms KarlF?WarnickandDavidV?Arnold ShortTitle?ElectromagneticGreenForms DepartmentofElectricalandComputerEngineering ?ClydeBuilding BrighamYoungUniversity Provo?UT? Warnick?etal?September? Abstract?InthispaperweredevelopthescalaranddyadicGreenfunctionsof
2、 electromagnetictheoryusingdi?erentialforms?TheGreendyadicbecomesadou? bleform?whichisadi?erentialforminonespacewithcoe?cientsthatareformsin anotherspace?oradi?erentialform?valuedform?Theresultspresentedherecorre? spondcloselywiththeusualdyadictreatment?butareclearerandmoreintuitive? Manyoftheusuale
3、xpressionsusinggreenfunctionsinvectornotationrequirea surfacenormal?withtheGreenformsthesurfacenormalisunnecessary?Weillus? tratetheformalismbycomputingscatteringfromarandomlyroughconducting surfaceandderivingtheGreenformforadielectrichalf?space?Wealsode?nethe interiorderivative?whichisequivalenttot
4、hecoderivativebutforaconstantmet? richasa computationalruledualtothatoftheexteriorderivativeandsimpli?es calculationincoordinates?Thisworkmakesavailablesomeofthetoolsthathave notyetbeenpresentedinthelanguageofdi?erentialformsbutareessentialin appliedelectromagnetics? Warnick?etal?September? ?INTRODU
5、CTION InthispaperwetreatGreenfunctionmethodsinelectromagnetic?EM?eldtheoryusing thecalculusofdi?erentialforms?Thecalculusofdi?erentialformshasbeenappliedtoEM theorybyDeschamps?Baldomir?Schleifer?Thirring?Burke?Bamberg? IngardenandJamio?lkowksi?Parrott?andothers? Severalauthorshaveadvocatedtheuseofth
6、ecalculusofformsinengineeringEMtheory? butsomeimportanttoolsforappliedproblemshavenotbeendeveloped?In?theauthors presentedarepresentationofEMboundaryconditionsusingdi?erentialforms?Inthis workwedevelopanothertoolwellsuitedforpracticaluse?theGreenforminthe? representation? AsproposedbyThirring ?theEM
7、Greenfunctionbecomesadoubleform?Double formsarede?nedbydeRhamin?Greenformsaretreatedinthemathematicsliterature ?see?anditsreferences?andThirringgivesthetime?dependentGreenformforelectro? dynamicsinMinkowskispacetime?OurGreenformhasthesamecomponentsastheGreen dyadicinKong?andthereforeiseasilyrelatedt
8、otheusualmethodsinappliedelectro? magnetics?WederiveexpressionsfortheGreendoubleformintermsofthescalarGreen function?theelectric?eldduetoasurfacecurrentdensityandtheStratton?Chuformula? Theuseofdi?erentialformsmakes theresultspresentedhereclearerincertainways thantheusualvectoranddyadictreatment?Ino
9、btainingexpressionsforobserved?elds intermsoftheGreenforms?theproductrulefortheexteriorderivativetakestheplaceof severalvectoridentities?Thismakesthederivationmuchcleaner?Thedyadicexpressionfor observed?eldsduetotangential?eldsalongthesurfaceofanobservationregionusingthe Greendyadicincludesasurfacen
10、ormal?WiththecorrespondingexpressionusingtheGreen form?thesurfacenormalisunnecessary? Inthispaperwealsode?netheinteriorderivative?which isequivalenttothestandard coderivative?butsimpli?escalculationsincoordinateswhenthemetricisconstant?The computationalrulewhichweproposeisdualtothatoftheexteriorderi
11、vative? InSec?wereviewoperationsonformsandtreatdoubleformsbrie?y?InSec?wesolve Maxwell?slawsofelectromagneticsintermsoftheGreendoubleformandthescalarGreen Warnick?etal?September? function?Finally?inSec?weillustratethemethodbycomputingscattered?eldsfroma roughconductingsurfaceandderivingtheGreenformf
12、oradielectrichalf?space?Thiswork showsthatthecalculusofdi?erentialformscanbeusedinallapplicationstowhichGreen functionsanddyadicsaresuited? ?DEFINITIONS Inthissectionwegivede?nitionsandnotationtobeusedinSec?toderivetheGreen forms?Wede?netheinteriorandexteriorderivatives?theinteriorandexteriorproduct
13、sand theLaplace?deRhamoperator?Double?formsarealsointroducedinthissection? ?Operators Theexteriorderivativedisde?nedin? andelsewhere?Itcanberepresentedformallyas d? ? ?x i dx i ? wherex ? ?x n arecoordinatesonann?dimensionalspaceandthesummationconventionis used?Theexteriorproduct?istheantisymmetrize
14、dtensorproduct?sothatdx i ?dx j ? ?dx j ?dx i anddx i ?dx i ?Oftenthewedgebetweendi?erentialsisdropped?thereisan impliedwedgebetweenthedi?erentialsintheintegrandofanymultipleintegral?Thepartial derivatives ? ?x i actonthecoe?cientsofaform?sothatinR ? ?d?fdx? ?f ?y dy?dx? ?f ?z dz?dx sincedx?dx?Wede?
15、netheinteriorderivativeintheeuclideanmetricsimilarly? d? ? ?x i dx i ? whereistheinteriorproductde?nedin?InR ? wehaved?fdx? ?f ?x sincedxdx? anddydx?dzdx? Theinteriorproductisde?nedtobethecontractionofavectorwithak?form?whichis atotallyantisymmetric? ? k ?tensor?Inthispaperweusetheeuclideanmetric?so
16、wecan extendthisde?nitiontotheinteriorproductofa?formandk?formeasilysincevectorsand ?formshavethesamecomponents?Theinteriorproductofdx i j anddx i ? ?dx i k iszero Warnick?etal?September? fori j notequaltoanyofi ? ?i k ?otherwiseitis? j? dx i ? ?dx i j? ?dx i j? ?dx i k for?j?k?Thus?by?theinteriorde
17、rivativeofaformiscomputedbymovingeach di?erentialinturntotheleftmostpositionbyalternatingthesignoftheformeachtimetwo di?erentialsareswapped?removingthatdi?erentialandtakingthecorrespondingpartial derivative? Theinteriorderivativeisequivalent ? uptoasigntothecoderivativede?nedin?and elsewhere? d ? k?
18、 ? ? d? wherekisthedegreeof?and?istheHodgestaroperator?InR ? withtheeuclideanmetric? ?dxdydz?dx?dy?dz?dy?dz?dx?dz?dx?dyand? ? ?Notethat theinteriorderivativecontainsthesign? k? naturally?Foranonconstantmetric?such aswouldariseincurvilinearcoordinates?replaces?asthede?nitionoftheinterior derivative?
19、Theinteriorderivativeiseasiertocomputewiththanthecoderivative?asillustratedby thefollowingexample?We?rstusethecoderivativeto?nd ?d?D ? dydz?D ? dzdx?D ? dxdy? ? ? ?x dx? ? ?y dy? ? ?z dz?D ? dx?D ? dy?D ? dz? ?D ?x dx?dx?D ?y dy?dx?D ?z dz?dx ?D ?x dx?dy?D ?y dy?dy?D ?z dz?dy ?D ?x dx?dz?D ?y dy?dz?
20、D ?z dz?dz? ?D ?z ?D ?y ?dx?D ?x ?D ?z ?dy?D ?y ?D ?x ?dz? Usingthede?nitionoftheinteriorderivativewecomputethesameresultimmediately? d?D ? dydz?D ? dzdx?D ? dxdy?D ?y dz?D ?z dy?D ?z dx?D ?x dz?D ?x dy?D ?y dx wherewehaveremovedeachdi?erentialinturn?afteritmovingtotheleftifnecessaryusing theantisym
21、metryoftheexteriorproduct?andtakenthecorrespondingpartialderivative? TheLaplace?deRhamoperator?is ?dd?dd ? Warnick?etal?September? whichisageneralizationofthevectoroperatorr ? ?Withtheeuclideanmetric?becomes ? i ? P j ? ? ?x ? j ? i wherethesubscriptiindexescomponentsof?On?forms?isequiv? alenttothee
22、uclideanvectoridentityr ? ?r?r?rr? ThegeneralizedStokestheoremis Z V d? Z ?V ? where?isap?formandVisap?dimensionalregionwith?Vasitsboundary?Also?the interiorproductoftwoarbitraryformsaandbsatis?es a b?b?a? where?istheHodgestaroperator? ?DoubleForms Adoubleform?isadi?erentialforminonespacewithcoe?cie
23、ntsthatareformsin anotherspace?ThedoubleformsthatwewilluseinthispaperareassociatedwithR ? ?R ? ? whereR ? istheobservationspaceandR ? ? isthesourcespace?Wewilluse?formvalued ?forms?ordouble?forms?whichcanbewritteningeneral G?G ? dxdx ? ?G ? dxdy ? ?G ? dxdz ? ?G ? dydx ? ?G ? dydy ? ?G ? dydz ? ?G ?
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