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－－
ＶＥＣＴＯＲＡＮＡＬＹＳ昭
L1PropertiesofVectorsandCoordinateSystems
Ordinarynumbersarccａｌｌｃｄｓｃｑｌｑｒｓ・Ｔｈｅｙｍａｙｂｅｒｅａｌｏｒｃｏｍｐｌｅｘｎｕｍｂｃｒｓ・In
contrasttoscalars，wehaveotherquantitiescalledDectors､Thesequantitiescom‐
binewitheachotherdiHbrcntlythanscalarsJnphysicstheyareusedtorepresent
objectsthathavebothmq9"imdeanddirectio",theprototypeofwhichisadisplace‐
ment，Mathematically，vectorsaresimplyquantitiesthatbehaveandcombineaccordingtothelbllowingrules、
１．Ｔｈｃｓｕｍｏｆｔｗｏｖｅｃｔｏｒｓｕａｎｄｖｉｓａｎｏｔhcrvector：ｕ＋ｖ＝ｗ､Ｔｈｅｓｕｍｉｓａ
ｃｏｍｍｕｔａｔｉｖｅｂｉｎａｒｙOperation；ｔｈａｔｉｓ,ｕ＋ｖ＝ｖ＋Ⅲ
２．Undersummation，theassociativelawholds,Forvectorsu,ｖ,ａｎｄｗ，
（ｕ＋v)＋ｗ＝ｕ＋(v＋ｗ)＝ｕ＋ｖ＋ｗ
３．Ａｎｙｖｅｃｔｏｒｃａｎｂｅ“multiplied，，byascalartoyieldanothervector・
Weshallrepresentvectorsgeometricallybydirectedlinesegments(i､e､,arrows)．
Themagnitudeofthevectorisproportionaltotbelengthofthelinesegment,and
thedirectionisgivenbytheorientationofthearrow-thatis,thcdirectioninwhich
itpoints,TherulestobeMlowedinperfbrmingthis(vector)additiongeometrically
arethese(seeFig.11):Onadiagramdrawntoscalelayoutthedisplacementvector
u;ｔｈｅｎｄｒａｗｖｗｉｔｈｉｔｓｔａｉｌａｔｔｈｅｈｅａｄｏｆｕ,anddrawalinefromthetailofutothe
headofvtoconstructthevectorｓｕｍｗ・Thisisadisplacementequivalentinlength
anddirectiontc1thcsuccessivcdisplacementsuandv､Thisproccdurecanbcgen‐
cralizedtoobtainthesumofanynumberofsuccessivedisplacemcnts．
1.1.1Ｂ日seVI3CtorsamdCoordinateSystems
ChoosingacoordinatesysteminspaceisessentiaIlyequivalenttochoosingasetof
basevectors・Ifwechooseacartesiancoordinatesystem(Fig.1.2),ourbasevectors
arechosenlLobealongthreelixedmutuaUyperpendicular(orthogonal)fixcddirec‐
tionscalledthex，ｙ，andzdirections・Ｉｆｗｅｒｅｐｒｅｓｅｎｔａｖｅｃｔｏｒｂｙａｎａｒｒｏｗ，ｔｈｅ
１
ZVECTORANALYSIS
鋤uureLJGeometricaldefinitionofthesum
oftwovectorsuandv．
Ｚ
ｙ
Ｘ
胸"｢B１．２Definitionofthecartesiancoordi‐
natesystem，showmgtheassociatedunit
vectors．
perpendicularprqjectionsofthearrowuponthethreecoordinateaxcsarecalledthe
cartesjq〃compo"e"tsofthevectorinthesedirections､Intermsofthesccomponcnts，
ｔｈｅｍagnitudeofavectorAisaslbllows*：
MagnitudeofA＝|Al＝い:十Ａ;＋Ａ:)uz
AunitvectorAisthatvectorwhichwhenmultiplicdbythemagmtudelAl,yields
thevcctorA;thatis,Ａ＝|AlAItprovidesameanslbrindicatingdirection・Unit
vectorsalongx,ｙ,andzcoordinateaxcs(cartesian)ａｒｅｄｅｎｏｔｅｄｂｙｋ,’'2,respec‐
tively､Theyprovideaconvenicntandlimdamentalsetofbasevectors・Intermsof
cartcsianunitvcctors,anyvectorAisrcpresentedby
A＝ｄｪjt＋Ａ,，＋Ａｚ２（11）
ｗｈｃｒｅＡ罪,４，，ａｎｄＡ葱arethecomponcntsofAalongtheA,，,and2directions，
respectively、
Weshallrestrictourattentiontocaseswherethebasevectorsfbrmanorthog-
onalset・Moreoverthemagnitudeofeachbasevectorwillbetakenasunity(orthonormal).ThemostcommonlyusedbasevectorsinordinarythrCe-dimensionalspacc
aretheunitbasevectors(n,'’2).ThesevectorsareconsideredtobecoFIsm"ＷＣＣ‐
tors・Neithertheirdircctionsnortheirmagnitudesdepcndonwhelctheyarelocated
withrespecttosomerelerencepointinspaceItistheconstancyofthisortho‐
ｎｏｒｍａｌｓｅｔｏｆｂａｓｅｖｅｃｔｏｒｓｔｈａｔｗｅｗｉｓｈｔｏｅｍｐｈａｓｉｚｅｂｙｔｈｅｗｏｒｄｃｑ汀esjq"．
＊ThroughOutthisbookscalarsareinitalicsandvcctorsaにinboldItlce．
1.1ＰＲＯＰＥＲＴＩＥＳＯＦＶＥＣＴＯＲＳＡＮＤＣＯＯＲＤＩＮＡTＥＳＹＳＴＥＭＳ３
TherepresentationofvectorsusinｇｔｈｅｕｎｉｔｖｅｃｔｏｒｓｉｓｖｅｒｙｕｓｅｌＵｌｉｎｖector
manipulations・Ｆｏｒexample，ｔｏａｄｄＡｔｏＢｗｅｓｉｍｐｌｙａｄｄｔｈｅｃａｒｔｅｓｉan
components：
Ａ＋Ｂ＝仏兵＋Ｂｪ)Ｒ＋い,＋Ｂ,),＋(Aご＋Ｂご)２
Itisfrequentlyconvenienttouseothcrsetsofbasevectorswhosedirectionsdo
happentodependontheirlocations(curvilinearbasevectors）Ｆｏｒexample,we
shalldelineandoftenuseasphericalcoordinatesystemandacylindricalcoordinate
system・ＦｏｒｅａｃｈｏｆｔｈｅｓｅｓｙｓｔｅｍｓｗｅｓｈａｌＩｆｉndanorthonormalsetofassociated
basevectorsthatdependonwhereinspacetheyarelocated
RecallthatthecartesianunitvectorRmaybedefinedastheunitvectorthatis
perpendiculartoanyplanex＝constantSimilarly,lbrtheland2unitvectorswe
respectivelyassociatetheplanesy＝constantan。ｚ＝constant・Ｎｏｗ，thereare
othersurfacesthatoncca、describethatcorrcspondtosomegeometricalvariable
beingconstant・Ifwecanfindthreesurlaces,definedby(three)geometricalvari‐
ables,thatintersecteachotherperpendicularlyatapoint,thenatthispointwecan
definethreeassociatedmutuaUypelpendicularvectorｓｔｈａｔａｒ己normaltothese
surfaces・Indescribingthesphericalandcylmdricalsystems,wecitetwoinstances
wherewelindituselilltodoso.(Therearemanyothcrs.）
Inthecylindricalcoordinatesystem(Fig.1.3)wedefineasetofbasevectorsata
pointbyconsideringsurfaces，ｔｗｏｏｆｗｈｉｃｈａｒｅｐｌａｎｅｓａｎｄｏｎｅｏｆｗｈｉchisa
cylinder･Thesurfacesaredenotedbythelbllowingequations：
（a）ｚ＝constant
〔I,）β＝constant＝､/〕ＦＴｱ，
（c）①＝constant＝tan-1(y/x）
Inequation(a)thezcoordmatespecifiesasetofparallelplanes・Itisdefinedby
relbrcncetoareferenceplanecaUedthez＝ＯｐｌａｎｅＴｈｅｕｎｉｔｖｅｃｔｏｒ２ｉｓｔｈｅｎａ
ｃｏｎstantvectorpointinginthe(positive)dircction(whichmaybechosenarbitrari‐
ly),perpendiculartothez＝constantplanes・Thezaxisischosentobealine
pointinginthezdirection(ｆｂｒ－ｏｏ＜ｚ＜＋ＣＯ)．
Ｚ
ソ
死
鞠H､℃１．３DefinitionofthecylindricaIcoordinate
system,showingtheassociatedmitvectors．
４ＶＥＣＴＯＲＡＮＡＬＹＳＩＳ
InCquation(b)thepcoordinateisdefincdwithreierencetothezaxisbyasetof
cylindricaIlycircularsurfaccsthatintersectthez＝constantplanesperpendicularly・
Thedistanccphomaparticularsurlａｃｅｔｏｔｈｅｚａｘｉｓｉｓｔｈｅｒａｄｉｕｓｏｆｔｈｅｃylindrical
surfaceTheunitvectorPisperpendiculartothecylindricalsurface,pointingaway
lromthczaxisJtsdirectiondependsuponwhatgeometricalpointofthecircle
resultingfromintersectionoftheplanesz＝constantandp＝constantiscon‐
sidered・Thus,ｉｎFig.13,Pisaiimctionofthevariableddefinedinequation(c）
(ｌｂｒｐ＞Ｏ)．
Inequation(c)theonlysurfacesthatcanperpendicularlyintersectthepreviously
definedsurfacesatallpointsofintersectionareplancsthatcontainthezaxis・Ｏｎｅ
▲
suchplaneiscallcdthe′＝OplancandischosenarbitrarUy､Ｔｈｅ○unitvectorslie
perpendiculartotheの＝constantsurfblces,ａｎｄdependupontheangleのofthe
planewithrelerencetotheの＝Oplane
Theintersectionofthesuriacesdescribedbyequations(a),(b),ａｎｄ(C)above
locatepointsinspacejustastheintcrsectionofthecartesiancoordinateplanesdo
However,thecylindricalunitvectorsarewellspecifiedonlywhenapoint(noton
thezaxis)isspccilicd.(Theoriginisspecifiedbysettingz＝Ｏａｎｄｐ＝0.)Oncethis
hasbcendoncbyassigningvaluesof(，,巾,z)｡r(x,y,z)tot1Jepoint,anyvcctormay
beexpressedintermsofthecylindricalunitvectors(2,0,巾)atthatpoint、
Onecaneasilyshowthatthesecylindricalunitvectorsarerelatedtothecartesian
unitvectorsbythefbllowingrclations．
Ａ
Ｐ==fcosの＋，sinの巾＝－Rsinj＋，cosj２＝２
（1.2）
jb
RcmcmbcrthatPand巾dependuponthecoordinaPj.Thus,IbranyvcctorAand
鯛.Wlhi;搬紺鯛i鰯聯嚇｡,i,綜繩槻紬:源
basevcctorsusedtoexpressAwillbethatdefinedbythelocationr・Notethatthe
displacementvectorTtoapoint(z,p,の)ｉｓｇｉｖｅｎｂｙｒ==叩＋z2．
Weshallnotdescribethesphericalcoordinatesystem(Fig.1.4)inthedetailused
abovelbrtlibcylindricalsystem,excepttonotethattheconstantsurlaceschosenare
asfbllows：
(a）
(b）
に）
r=constant=､/;5ZT7ZT三whichdescribcsasphcreofradiusrwith
respecttotheorigin
O＝constant＝ＣＯＳ~'(z/rLwhichdescribesarightcircularconewithopen‐
ｉｎｇａｎｇｌｅａ
ｄ＝constant＝tan~'(y/X),whichdescribesaplanecontainingｔｈｅａｘｉｓｏｆ
ｔｈｅｃｏｎｅｉｎ(b)．
Theunitvcct･rsprescribedbythcsesurlacesaredcnotedf,04,respectivcly,and
fbrmanorthonormalsetoncethepoint(notattheorigmandnotatthezaxis）
locatedbytheintcrsectionofthethleeorthogonalsurfacesisdeterminedThese
unitvectorsarcgivenintermsofthecartesianunitvcctorsbythefbllowing
relations．
ｆ＝父sinOcos中＋jsinOsinj＋zcosO
▲
Ｏ＝父cosOcosの＋ｊｃｏｓＯｓｉｎｄ－２ｓｉｎＯ
６＝－ksin中＋，COS巾
(1.3）
ＬｌＰＲＯＰＥＲＴＩＥＳＯＦＶＥＣＴＯＲＳＡＮＤＣＯＯＲＤＩＮＡＴＥＳＹＳＴＥＭＳ５
ｂ
Ｚ
｣〉
ｙ
Ｘ
胸"r２１.‘Delinitionofthesphericalcoordinatesystem，
showingtheassociateduniWectors．
IfAisamemberofavectorlield,Ａ(r),thenatevcrypointgivenbythedisplace‐
mcntronccan目xpressA(r)intcrmsofthebasevectorsassociatedwiththatpoint：
A(r)＝Ａｒf＋Ａ００＋Ａ中中,ｗｈｅｒｅＡ，ｉｓｔｈｅｐｒｑｊｅｃｔｉｏｎｏｆＡｏｎＡａｎｄｓｏｏｎ､Thc
displacementvectortoapoint(｢,0,．)isgivensimplyr＝Ｇ７、
1.1.2TheScalarProduct(DotProduct）
Animportantconceptinvectoralgebraisthatofthescalarproductoftwovectors・
ＩｔｉｓｄｅｎｏｔｅｄｂｙＡ・Ｂａｎｄａｌｓｏｃａｌｌｅｄ‘otprod"ｃｔｏｒｉ""erprod皿α、Itisdefined
accordingtotheIbllowingruIe：Ａ･Ｂ＝|AllBlcoso上,ｗｈｅｒｅｌＡｌａｎｄｌＢｌａｒｅｔｈｃ
ｍａｇｎｉｔｕｄｅｓｏｆＡａｎｄＢ,ａｎｄｏＣｉｓｔｈｅａｎｇｌｅｂｅｔｗｅｅｎＡａｎｄＢ・Itcaneasilybeseen
thatthescalarproduct,asdefined,hasthelbllowingproperties:Twovectorswhose
scalarproductiszeroaresaidtobeorthogonal;ｔｈａｔis,ｉｆＡＢ＝0,Aissaidtobe
orthogonaltoBTheunitcartesianvectorsR,，,and2arcsaidtoconstitutean
o〃ｈｏ"０ｍｍlsetofbasevectorsbecausetheyareorthogonaltoeachotherandtheir
magnitudesarenormalizedtounity．
1.1.3TY1eVectorPmduct(CrossProdUct）
ＷｅｈａｖｅｓｅｅｎｔｈａｔｗｅｃａｎａｓｓｉｇｎａｓｃａｌartoanypairofvectorsTheoperationthat
doesthisiscalledthescalarproduct・Wenowwishtoassignavectorquantityto
anypairofvectors,ＡａｎｄＢ,andsowedefinewhatisknownasaDectorprodtJct(or
crossprod脚α);itisde､otedbyAxBThedirectionofthevectorproductistaken
tobeperpendiculartotheplanedeterminedbythepairofvectorsItsmagnitudeis
givenbytheareaoftheparallelogramwhosesidesarefbrmedbythevectorpalr・
Therelbre,ifhisaunitvectorperpendiculartｏｔｈｅｐｌａｎｅｌｂｒｍｅｄｂｙｔｈｅｖｅｃｔｏｒｐａｉｒ
(A,Ｂ),thenthcvectorproductisde6nedaccordingtothelbUowingrule
AxB＝ｌＡｌｌＢｌｓｉｎａｉｉ（1.4）
６ＶＥＣｒＯＲＡＮＡＬＹＳＩｓ
霧
JYU"｢Ｆ１・sDefinitionoftheright-handscrew
convention,ｗｈｉｃｈｇｉｖｅｓｔｈｅｓｅｎｓｅｏｆｔｈｅｃｒｏｓｓ
ｐｒｏｄｕｃｔｏｆｔｈｅｖｅｃｔｏｒｓＡａｎｄＢ．
SofarthedefinitiongivenremainsambiguousbecausethenormaltotheplancofA
andBmaypoint“up，，ｏｒ“down."Tospccilywhichwayiipoints,weusetherjght‐
hα"ｄｓｃｒｅｗｃｏ"DC"tjoPl､WesaythatifAisrotatedtothCdirectionofB,throughthe
angleaに180｡),ｔｈｅｎｔｈｅｓａｍｃｓｅｎｃｅｏｆｒｏｔａｔｉｏｎｇｉｖｅｎｔｏａright-handedscrew
determinesii，whichpointsalongthedirectionofａｄｖａｎｃｅｏｆｔｈｅｓｃｒｅｗａｓｉｔｉｓ
ｒｏｔａｔｅｄ(Fig.１５l
IntelTnsofcartesianumtvectorsｔｈｅvectorproductisexpressedas
AxB＝jt[Aj,B霊－Ａ二B，]＋，[AzBx-AxBz]＋2似蒸By-AyBz］（L5）
，４酌
父上瓜
､ＡｘＢ＝
鮴…‘
(1.6）
Wheneverwehavcasctofthreeorthonormalvectors,、１，６２，ａｎｄｅ３，ｗｅｓａｙｗｅ
ｈａｖｃａ'i9ht-hα"dedsystemwhen6ix§＝Gkwhcrej,ｊ,kareintheorder(1,2,3)，
(2,3,1),ｏｒ(3,1,2).ThesearecyclicpermutationsoftheintegCrsl,２，３.Notethat
fbraright-handedsystemgivenbythctriplet{6,,62,63},onehase1･(e2xe3)＝L
Thecartcsiancoordinatesystemwehaveusedisright-handedifweidentilyxwith
l,Jﾉｗｉｔｈ２，ａｎｄｚｗｉｔｈ３､Ａｌq/t-liqndedsystemisamirrorimagCofaright-handed
systCm・
TWousefUlidentitiestobcrcmcmberedareaslbllows：
１．Triplcscalarproduct
A･(ＢｘＣ)＝(ＡｘＢ)｡Ｃ＝(ＣｘＡ)･Ｂ（1.7）
（ItisthepollJmeofaparallelepipedwhoseedgesareA,Ｂ,ａｎｄＣ）
２．Triplevectorproduct
Ax(ＢｘＣ)＝Ｂ(ＡＣ)－Ｃ(Ａ･Ｂ）（1.8）
Thelatterlisficqucntlyknownasthe“backcab”rule､ItwillbenotedinEq.(1.7）
thatthcdot(･)andthecross（×)maybefrcclyimerchangedsolongas{A,Ｂ,Ｃ）
rcmainincyclicordcr．
L2ElementsofDisplacement,Area，andVolume；SolidAngle
1.2.1ElementofDiSplacement
Considertwopointsinspace(x,y,x)and(ｘ＋△x,ｙ＋△y'２＋Ａ２).Thefirstpointis
displacedrelativetothesecondbythcdisplacemcntＡｒ；ｔｈａｔｉｓ，
Ａｒ＝ｊｔＡｘ＋，△ｙ＋２Ａ２（19）
L2ELEMENTsoFDIsPLAcEMENT,AREA,ＡＮＤｖｏＬｕＭＥ；ｓｏＬＩＤＡＮＧＬＥ７
Ｚ
ｙ
Ｘ
胸ElreJ.‘Thecoordinatesoftwonearbypointsin
cylindricalcoordinatesthatmaybeusedtodefinethedilL
lerCntialdisplacementsinthissystem
expressedinacartesiansystemAdiflbrcntialelementofdisplacementisconscquentlywritten
‘r＝ｄｘＲ＋ｄｙｌ＋dz2
（1.10）
WenowwishtoexprcssAr(ordr)intermsofcylindricalandsphcricalcoordinatcs
andtheirassociatedunitvectors､ＷｅａｇａｉｎａｓｓｕｍｅｔｈａｔＡｒｍａｙｂｅｍａｄｅａrbitrarily
small,ｉｎｔｈｅｌｉｍｉｔｃａｌｌｉｎｇｉｔｄｒ・
considerFig.1.6,wheretwopointsarしdisplacedbyAr・InthecylindricalcoordinatcsystemwchavebasevectorsthatarcdilＹｂｒｃｎｔａｔｔｈｅｔｗｏｐｏｉｎｔｓｌａｎｄ２Ｔｈｕｓ
Ａｒ＝p2p2＋2222-(ハハ＋ｚ12,）（1.11）
where
Ｐｚ＝ｐ,＋ＡＰＰ2＝ハ＋ＡＰＺ２＝z,＋Ａｚ２ｚ＝２，＝２（1.12）
SubstitutingEq.(1.12)ｉｎＥｑ.(1.11)anddroppingproductsofdifYbrcntials,weobtain
Ar＝Ａｐハ＋ｐ１Ａｐ十Ａ２２
（113）
Ifpoints(1)ａｎｄ(2)areclosecnoughtogetherthen,togoodapproximation，
ﾑﾄ鶉帥
ForApsufYicientlysmall(6,-62＝⑤wccansecthatAp＝|ハｌＡｊｉ・ＡｓａEcsult，
Eq.(1.12)becomes
Ａｒ＝Ａｐハ＋ｐ,△｡,＋Ａｚ２
（1.14）
Aspoint2approachespointl，wecanwritethedifYerentialdisplacementas
‘r＝dlbp＋dﾉ`巾十.【’
（1.15）
dl,＝ｄｐｄ４＝ｐｄｄｄＬ＝｡ｚ
（1.16）
whelc
aretheelcmentsofdisplacemcntinthep,の,andzdircctions,respcctively・Ｔｈｕｓ
‘lr＝ｄｐｐ＋，｡の‘＋dz2
（1.17）
８vEcroRANALYsIs
zＨｌ
皿ａｎ
吻醒１．７ＤiHerentialdisplacementsinspher‐
O
icalcoordinates．
whereいゆ,z}arcdelinedatthepointwhcrethedisplacementdrismade・ＧＣＣ‐
metricallythisisanaturalresult,sinceclosecnoughtoanypointofspace(２，，，①）
ｗｅｃａｎｄｅｆｉｎｅａ“cartesianlike，，systemintermsofwhichanyelementoflengthmay
beexpresseddirectly
Insteadofderivingtheelcmentofdisplaccmenｔａｔａｐｏｉｎｔｉｎｔｅｒｍｓｏｆｓｐｈｅｒｉｃａｌ
ｕｎｉｔｖｅｃｔｏｒｓａｔｔｈａｔｐｏｍｔ,ｗｅｓｈａＵｓｉｍｐｌｙｍａｋｅｔｈｅｒｅｍａｒｋｔｈａｔａtanypointthe
unitvcctorsM,6}servetoestablishacartcsiansystemlocaUy(nearthepoint)Ｉｔ
ｗｉＵｂｅｓｅｃｎｆｒｏｍＦｉｇ・L7thatthcclementsoflengthalongthethreedircctionsncar
thispoinｔａｒｅｇｉｖｅｎｂｙ
‘し＝`「。ｌ０＝ｒ`０．１０＝ｒｓｉｎＯｄの（L18）
ａｎｄｔｈａｔ
へ
。『＝drf＋,dOO＋,sinOd仰＝｡！『f＋d４０＋`！`巾（119）
へ
1.2.2EIementofSurfaceArea
Havingdeterminedexpressionsfbrelementsofdisplacemcntinvanouscoordinate
systems,wecannowdetermineclementsofsurfacearea・Thcrcarethrceclementsof
surfaceareafbrcverycoordinatesystem;theseareofthefbrm‘Ｉ,〃２，ｄｌ２‘13,and
dl3dl1・Forcartesiancoordinates,ｗｃｈａｖｅ
ｄｘｄｙｄｙｄｚｄｚｄｘ
（1.20）
correspondingtothesurfacesz＝constant,ｘ＝constant,ａｎｄｙ＝constant,respec-
tively･Similarly,fbrcylindricalcoordinates,clementsofsurfaceareaonthesurfaces
thatdefinethecoordinatesarｅ
ｄｚｄｐｐｄｐ⑭ｐｄの｡2
0.21）
Forsphericalcoordinateswehave
wlrdOr2sinOdO`のrsinOdj力（1.22）
Adirectionmaybeassociatedwithanclementofarea・Thisdirectionisnormalto
thearea･IfdJidぃstheclemcntofarea,thenormaldirectionisgivenbythccross
productejxq,andwemaydenotetheareaasaVCctor(6,.4)×(`川)＝Ｍｌｌｄｌｊ．
l2ELEMENTSOFDISPLACEMENT,AREA・ＡＮＤＶＯＬＵＭＥ;ＳＯＬＩＤＡＮＧＬＥ９
０
D゛
〃、
￣
伽
Ｚ
＝、
｡
３㎡Bａｄ
ｙ
Ｘ
（６）
胸luyごＩ､８DehnitionofplaneandsoIidangles.(α)P1arueangIedO
(6)SolidangledQ．
Olienweshallmoresimplydenotethｅｅｌｅｍｃｎｔｏｆａｒｅａａｓｄａ＝ｈ‘α,wherethesensc
ofnmustbespecifieｄｂｕｔｉｓａｌｗａｙｓｎｏｒｍａｌｔｏｔｈｅｓｕｒｆａｃｅ．
1.2.3SOlidAungle
Whenanarcelementdｓｏｆａｃｉｒｃｌｅｉｎａｐｌａｎｅｉｓｒｅｉｅｒｒｅｄｔｏitscenter,weusethc
conceptofanangledO＝CMwherCristheradiusofthecircle(Fig.18α).Onthc
otherhandwhenanelementofsurfaceareaisrefbrredtoanorigin,itjsolten
convenicnttousetheconccptofsoljdan此(seeFig.1.86).ThediHbrmtialelement
dQofsolidanglcwithrespecttotheoriginisdefinedaslbllows：
…等=竿=型
ｒ２
(1.23）
Ｈｅｒｅ,thesurfaceelementdtJislocatedatapointdisplacedfromtheoriginbythe
vectorr＝fr,ａｎｄｈｅｎｃｅγistheanglebetweenOandfSinccda･fisjustthe
elementofareaofasphereofradiusr,thensubstitutingdq＝r2sinOdOd中,ｗｅｓｅｃ
ｔｈａｔｄＱｉｓａｌｓｏｇｉｖｅｎｂｙ
ｄＱ－血誓｡藤｡-'２sin8`6`の=sinodOdのＭ）
ｒｚ
ｗｈｉｃｈｉｓａｎｅｌｅｍｅｎｔｏｆａｒｅａｏｆａｕｎｉtsphere
Physically,thesolidangleisthe“openingangle”ofaconewhosesidesintercept
theareaelemcntinqucstionThus,justasfbrordmaryangularelementsdO,wherc
l０ＶＥＣＴＯＲＡＮＡＬＹＳＩＳ
〈、
〈、
'
、
〈、
】Ｉ
３
０
胸”e1.9111ustratingwhythesolidangle
subtendedbyaclosedsurlacewithrespectto
anorigininsidethesurklceis4兀andtoan
originoutsidethesurlaceiszero．
ｗｅｈａｖｅｒ`O＝ｄｌｃ…＝小Csし,fbraninfinitesimalelementofarcａｏｆａｓｐｈｅｒｅ
(whichapproximatesarectangularplanararea)ｗｅｈａｖｅｄａｓｐｈｃｒｃ＝r2sinOdOd①,。α
＝ｒ２ｄＱ､Ｔｈｅｕｎｉｔｏｆｓｏｌｉｄａｎｇｌｅｉｓｋｎｏｗｎａｓｔｈesterqdmnltisclearlyanalogous
ｔｏｔｈｅｕｎｉｔｏｆａｎｇｌｅ，theradianAnylinitesolidangleisexpresscdas
Q＝IdQ＝IsinOdOdj,whereOand①arethesphericalcoordinatesofthcspher‐
icalsurfaceelementinterceptedIfwehaveasurfacethatcompletelyenclosesthe
origin,ｔｈｅｎＱ＝４兀.Ifwehaveaclosedsurlacethatdoesnotenclosetheoriginand
ifwechoosethedirectionofdaalwaystopointoutlromthecloscdsurlace(orinto
thcsuriacc),thenＱ＝OThisis(seeFig.1.9)essentiallyduetothelactthatibreach
positivecontributionofsolidanglethereisanequalcontributionofnegativesolid
angle,asseenlromtheorigin
1.2.4ＥIementofVoIume
RememberingthatA･(ＢｘＣ)isthevolumeofaparallelepiped,wehavethatthe
volumeelcmentlbrasystcmofbasevectors{6,,62,,3}issimplygivenbythevcctor
tripleproduct：
‘､＝ｄﾉ,、,．(｡！２６２×,ﾉﾉ363）
ｏｒ
‘､＝'６，.(ｅ２ｘｅ３)|ｄｌ,。ﾉｭdl3
wherethed↓arcthemagnitudcsoftheelcmentsofdisplaccmcntsalongthedirec‐
tionofthercspcctivcbasevcctors§・Thus,inthecartcsiansystcm，
。U＝'2.(，×２)Mxdydz＝dxdlydz
（1.25）
Inthecylindricalsystemwehave
へ
。､＝'2･(ｐｘ０)ldzdpp⑭＝pd2dpdj（126）
andinthesphcricalcoordinatesysｔｃｍｗｅｈａｖｅ
▲へ
。u＝'6．(Ｏ×｡)|`rrdOrsinOd①＝r2sinOdrdO⑭（1.27）
L2ELEMENTSOFDISPLACEMENT,AREA,ＡＮＤＶＯＬＵＭＥ;ＳＯＬＩＤＡＮＧＬＥ１１
１．３Ｇradient
lfwewisbtoexprcssthechangeinascalarfUnctionofposition／(r)atthelocation
spCcifiedbyr,thenwriting/andrincartesiancomponents,wenndthediHerential
changetobc
…襄十`,静雄墓
WenowdefinealineardilYerentiaI“vectoroperator，，caUcddel,andsymboIizedV
asfbllows：
，臺徽急十，急十m是
(1.28）
‘＝(WMr
(1.29）
Sincedr＝Ｒｄｘ＋，｡〕＋２．z,thenlromthedefinitionofthedotproduct，
whercWisavectorpointfimctionjandiscalledthegradie"ｔｑノノ：
狐にw-捌癸+，筌十2壬
(1.30）
Someapplicationsinvolvetheoperationoftheabovegradientoperatoronvector
ficldslfwewishtoexpressthechangeinavectoriieldAatthelocationspecifiedby
r,thenwritingAandrincartesiancomponents,wefindthediHbrentialchangetobe
‘…姜十`,警十`帯[`蕊云十`,鼻+`釘△
UsingEq.(1.30LonecanshowthatthcscalarproductofdrandVis
Thus
‘w-["呈川,急+`塾呈］
(131）
｡Ａ＝(drV)A(r）
(1.32）
Inwords,thescalaroperator(dr.Ｖ)actingonavectorpointhmctionA(r)generates
thespatialdifYerentiaｌｏｆＡ,。Ａ,atthepointinquestion．
０
c＋〃
胸耀1.IOTheuseofthesmface（『)＝ｃ＝constanttoshow
thatthegradientofthelimction/(r)isnormaltothissurface．
１２vEcroRANALYSIs
ThegradientgiveninEq.(1.28)hasanmterestinginterpretation(seeFig.1.10)．
SupposcwehavethescalarfUnction/(r).Asurlnceisgeneratedifweset/(r)equaltoa
co"stqFDt,c・Thisbeingthccase,ifwefindthedifYerentialof/(r)whendrconnectstwo
pointsoftheSurface/(r)＝c,then(callingdいhissurfacedisplacement)/(r＋`⑲＝
/(r)＝C,ｓｏthat〃＝OThisimpliesthat‘＝(Ｗ)･`r図＝0,whichinturnim‐
pliesthat(Ｗ)isperpendiculartodr,､Sincedrswasdelinedtolieinthetangentplane
tothesurfaceatr,wesCethatWisperpeJMic卿lqrtothesurfacc､Nowthevalueofthc
changCin/(r)whenwcmovetoancighboringpoint(r＋伽)notinthesurlacc,isgiven
byEq.(1.29):ガーＷ･`r・Therefbre,ifldrl＝｡s,〃/as＝(WMVlis＝Ｗ･i,ｗherei
isaunitvectorinthedirectionofdr・Thederivatived//【lsisknownasthedirectio"ql
dlerjDatipeofメThemaximumvalueoMWlswillbeobtainedwheniliesalongWIn
thiscaselW･il＝|Ｗ|＝〃/ds…;thatis,themaximumratcofincrcascinthe
lilnctionノ(r)withrcspecttodisplacementsdTisgivenbyＷ;andWpointsintbe
dircctionofmaximumincreasaThesepropcrtiesofWaresummarizedbytheIbrmula
w-m藍
(1.33）
whcreiid"isthcdisplaccmentdrperpendiculartothesurlaceノーconstant(mthe
directionofthemaximumincreaseinDItisfbrthisrcasonthatitiscalledthegradient
ofメAsanexample,consider／＝ｘ＝c,whichdefinesasetofplancs,onefbreach
valueofc,perpendiculartothexaxis・clearly,thelactthatＷ＝jtshowsthat(a)the
perpendiculartotheplanesliesinthexdirection,and(b)therateofincreaseof/in
thisdirectionisunity,Asanotherexample,letノーｘ２＋y2＋z2＝７２.Thenonefinds
W＝276,whereｒ＝ｘｉ＋ｙ，＋z2Theperpendiculartothesurlacesノーcareinthe
directionsoftheradiitothepointsonthespheresgivenbyxz＋ｙ２＋２２＝７２.Therate
ofchangeof/inthesedircctionsisjustd/Wlr＝27.Finally,notethatsinceWis
perpendiculartothesurfaceノーconstant,itcanbcusedtogeneratetheunitvectors
ofcoordinatesystcmsThus
－－
ｉ
ｖｌＭ
(1.34）
isaunitvectorpcrpendiculartothesurlace/(x,y,z)＝constant、
WenowiindanexpressionfbrWwhere/isanyscalarpointhmctionintermsof
cylindricalcoordinates・Todothisweturntoourdefiningequationfbrthegradient：
▲
`ｒＷ＝`lfWritmgdr＝Pdp＋ipdの＋2`lzandW＝P(Ｗ),＋巾(Ｗ〃＋2(Ｗ)塁，
aswecanalwaysdoatanypoint{，,の,z},giveｓ
"=(Ｗ),｡'+(Ｗｂｐｄｊ+(Ｗ)zdz
Notingthattherulesofcalculusrequire
炸莞`,+莞"+筌傘
andequatingthetwoexpressionsfbr〃gives
腓嘉（wﾙｰ潟（wﾙｰ美
ｏｒ
w-,差十＄鵠+鶚
Ｌ４ＴＨＥＤＩｖＥＲＧＥＮｃＥｏＦＡｖＥｃｒｏＲＡＮＤＧＡｕss，ＴＨＥｏＲＥＭI3
Hence,thcdel(V)operatormcylindricalcoordmatesiｓ
’一,完+`誌十塾呈川
Similarly,onecanobtaincxpressionslbrWandVinsphericalpolarcoordinates：
可一助０｜〃
ｌ｜》１｜“
〈抑へⅦ
可一”６｜卵
十＋
ｌｌｒｌｌｒ
Ａ０へ０
〃一介６房
｜’｜｜
▽▽
(1.36）
1.4TheDivergenceofaVectorandGauss,Theorem
Foravectorfield,Ａ(r),wcshalldelinethedivergence,writtenvariouslyasdivAor
V．Ａ，bytheCxpression
,A一隅[半］m
whercAレisadiHbrentialvolumeandSisitsSurface､MoreoverV・Amaybeshown
tobeequivalentlydefinableasthcscalarproductofVwiｔｈｔｈｅｖｅｃｔｏｒＡ・WewiU
firstusetheintegraldelinitiontoderivethedivergenceofavectorcxplicitly・ConsiderasurflceSwhosesurlaceelementisdenotedbWla,ｔhentheintegral
’一L…川
iscalledthe`yh4xofAthroughS.”clearly,asinourdiscussionofsolidangle,Sis
assumedtohaveanonentation(withrespecttotheoriginofthecoordinatesystem
used),andingcneralmayormaynotbeclosedTheelementofUuxthroughdais
correspondinglygivenbydF＝Ａ・dawhereAisevaluatedatthecenterｏＭａ・Now
considerthcinfinitesimalvolumeelementdp,ｓｈｏｗｎｉｎＦｉｇ､1.11.ＴｈｅｓｉｄｅｓｏｆｄＤａｒｅ
ｇｉｖｅｎｂｙｔｈｅｓｉｘｓｐherical-coordinatesurlaces:「＝ｃ,「＝ｃ＋ｄ｢;０＝ｃ'’０＝ｃ'＋ｄｏ，
(6)
〉
FiD…１．１１Determinationofthedivergenceofａｖｅｃ‐
torinsphericalcoordinates．
１４ＶＥＣｒＯＲＡＮＡＬＹＳＩＳ
の＝c",の＝c"＋ｄの,anditsvolumeis｢ZsinOdrdOd①.WecomputethefluxesdF1,
.Ｆ2,…，ｕｌＦ６ｔｈｒｏｕｇｈｔｈｅｓｉｄｃｓｌ，２，…，６０fthevolumeelementdcscribedabovｅ
(andUlustratedintheligure)．
｡F,＋dF2＝－(｢2A,sinOdOdj),＋(r2ArsinOdOd‘)2＝｡[r2sinOd8dのＡ』2,,
.F3＋dF4＝－(rsinOdのdrAo}3＋(rsinOd①drAルー｡(rsinOdjdrd`}4,3
.F5＋`ＩＦ`＝－{rdOdrA小＋(rdOdrA中}`＝｡(rdOdrAや)６，５
Thequantiticsinthebracescanbeevaluatcdatthecenterofthcfaces､Thus,lbr
surlacesland2onlylimctionsofrarediflbrent：
‘(んin",`“Ｍ_`(r2win，`,`トー(『塾4}…`M
Similarly，
‘(『Sim`…)…`(圏ｉｎＭＭ岬晶(sinO』`Ｍ`『`‘
Therdbre
‘(MM…-念い,Ｍ`M
研A薑ト鵲鶚`,］
thatｉｓ，
。A÷烏(柵赤[烏(…)十割（L，，）
WenowusethedirectscalarproductＶ･Ａｔｏａｒｒｉｖｅａｔｔｈｅｄｉｖｅｒｇｅｎｃｅｉｎｔｈｅ
ｖariouscoordinatesystems､TheVoperatorinsphericalcoordinatesisgivenbyEq．
(1.36),ａｎｄthus
…[，多半:烏+論念]Ｍ(｡+"］
Expandingtheindicatedscalarproduct,ｗｅｇｃｔｎｉｎｅｔｃｒｍｓ：
Ａ
▽乱→呈[……咄鵠[M…十帖］
Ａ
Ａ
ａＬ
Ａ
＋鈴急Ｍ…Ａｊ
Ｊｈ
Ｎｏｗ,perfbrmtheilnpliedderivatives,notmg,fbrcxample,ｔｈａｔ
２ドル筌峨'等
6７
andsofbrth・ThcrewiUbel8individualterms，ｂｕｔｉｆｗｅｒｅｍｅｍｂｅｒｔｈａｔ
６．０＝f‘＝０６＝0,andsoon,weobtainjustl2nonzeroterms：
Ａ
Ａ
ｖ…二Ａ,+等+臘豐…祭4．
十二諾4+鴻砧鶚十鵠糺，
+鈴覺4+論景朏鈴慕い元而而
▲Ａ
１Ｍ①
1.4ＴＨＥＤＩＶＥＲＧＥＮＣＥＯＦＡＶＥＣＴＯＲＡＮＤＧＡＵＳＳ，ＴＨＥＯＲＥＭ１５
ZiJ6化１．Ｊ
６
Ａ
０
Ｏ
Ｕ房６｜”６｜姉
0
0
▲
０
▲
－６
中sinO
0
`cosO－[OsinO＋Ocosm
濡搬i§,，樵離篭嚇鵬臘嚇瞬蝋纈柵鯏霊
glvenprcviouslymEq.（1.3).Ｆｏｒexample,sinceＯ＝kcosOcosd＋jcosOsin①
－２sinO､andthecartesianunitvectorsareconstanMO/60＝R(-sinO)com
＋,(－Sim)sｉｎ①－２cosO＝－G.Substitutionwillyicldthesameresultwegotin
L39・
TheoperatorVincylindricalcoordinateswasshowntobc［seeEq.(1.35)］
v－，急十$鵠+2基
ProceediPl9bjﾉｔｈｅｄｊｒｅｃｔｐＪ･ｏｄＨｃｔｍｅｔｈｏｄｔｏｃｑｌ“ｌａｔｅＶ・Aincylindricalcoordinates
andusingthefactthat2isaconstantunitvectorandthatOand中dependonthe
▲
coordinateのｏｎｌｙ［seeEq.(1.2)],wefindthattheonlynonzeroderivativcofthe
unitvectorsare6P/6①＝６and6巾/6の＝－P・Thcrefbre
wA-急い:』け:急峨筈
ｏｒ
…蒜(",)+誌峨筈
(1.40）
Finany,inthecaseofcartesiancoordinatestheoperatorVwasshowninEq.(1.28）
ｔｏｂｅ
v-m呈十'二十2基
ExpandingV・Adircctlygives
wA-等+等等
(1.41）
Z1beDilwge"CCZ】b巴”emrQzuuss，Z1bBorEum）．Finallyweprovearelationthatis
veryusefUlinelectrostatics：thedivergcncetheorcm,whichinvoIvesthedivergence
operationFromourdelinitionsofV・Ａｗｅｈａｖｅ
ａ
口α
Ａ
ｒ‐、Ｔ血
(Ｖ･Ａ)△､＝
１６VECTORANALYSIS
'１
si，△zIi
S，し
ノ
鋤"℃１.J2Provingthedivergencetheorem(Gauss，
law)bysubdividingthevolumeyofsurlaccSinto
manyinlinitesimalvolumcsADlofsurlacessiandthe
applicationofthedefinitionofthedivergenceinterms
oftheinlinitesimalvolumes．
●
lbrsmallAp､Consideravolumel′withasurfaceSsubdividedintoNsmallvolume
elementsVDi，ｗｉｔｈｓｕｒｌａｃｅｓｓｉａｓｓｈｏｗｎｉｎＦｉ9.1.12.Ｔｈｅｎ
ｚＩ,Ａｗ弓菫f…
ＩｆｗｅｔａｋｅｔｈｃＩｉｍｉｔａｓＮ→ｏｏａｎｄＡＤｉ→０，ｏｎｔｈｅｌｅＩｔｗｅｈａｖｅａｖｏｌｕｍｅｉｎｔｅｇｒａｌ
andontherightasurlaceintegral
Thelactthat
〃A`,-仏。川
JimLP妙。
becomesanimegraloverthesurfaceextemaｌｔｏｔｈｅｗｈｏｌｅｖｏｌｕｍｃｃａｎｂｅｓｅｃｎｌｒｏｍ
Fig.1.12.NotethatthenetHuxthroughanyintemalsurfaceiszero,ｂｃｃａｕｓｅｓｕｃｈａ
ｓｕｒｌａｃｅｉｓｃｏｍｍｏｎｔｏｔwocontiguousvolumeelementswhoseoutwardnormals
pointoppositelyｏｎｔｈｅｃｏｍｍｏｎｓｕｒｆａｃｅ・ＡｌｌｔｈａｔｒｅｍａｉｎｓａｒｅＨｕｘｅｓthroughsur‐
ｉｂＩｃｃｓｎｏｔｃｏｍｍｏｎｔｏｔｗｏｖｏｌｕｍｅｅｌｅｍents-namcly,surlhlcesonthesurfaccofy
itself
Equation（1.42LwhichisknownasthediDeP･ge"cetheorem(ｏｒＧａ卿ss，theorem)，
provesveryusehllwhenonewishestorelatevaluesofavectoriieldontｈｅｓｕｒｆａｃｅ
ｏｆａｒｅｇｉｏｎｔｏｖａｌｕｅｓｉｎｔｈeinterior・ＩｔｏｌｔｅｎｗｉＵｈａｐｐｅｎｔｈａｔｗｅｍａｙｗａｎｔｔo
convertasurlaceintegraltoavoIumeintegral，orviceversa，ａｓｗｉｌｌｂｅｓｈｏｗｎｉｎ
Ｃｈａｐｔｅｒｓ３ａｎｄ４．
L5TheCurlandStokes，Theorem
WenowintroduceanotherusclUloperationmvolvingV・Foravectorfield,Ａ(r),we
shalldelinethccurlwrittenasVｘＡｏｒｃｕｒｌＡｂｙｔｈｅｅｘｐｒｅｓｓｉｏｎ
ＶｘＡ＝
隅戯`…］
(1.43）
1.5ＴＨＥｃｕＲＬＡＮＤｓＴｏＫＥｓＴＨＥｏＲＥＭ１７
Ｚ
ｙ
X
Fi〃…１.J3Determinationoftheexplicit
lbｒｍｏｆｔｈｅｃｕｒｌｏｆａｖｅｃｔｏｒｉｎｃａｒｔｅｓｉａnco-
ordinatesusingadillbrentiaIrectangularbox．
ｗｈｅｒｅＡｕｉｓａｓｍａＵｖｏｌｕｍｅｏｆｓｕｒｆａｃｅareas,Thisdefinitioncannowbeusedto
determineVxAinacartesianrepresentation、Considerthevolumcclcment
dp＝dxdydzshowninFig､1.13.Thecontributionslromtheoppositesidesland2
totheintegralldaxAａｒｅ
ｌ"…(-M,…－…,－，幼
Ｌ`…-…ｘＡＩ－`MM，軌
Thesumofthesetwointegralsisthe(partial)difYbrentialoftheexpressionshownin
bracestakenbetwcenfacesland2：
Ｌ`…-…ル拠籔)止加`急(2,ＷMェ
-`假一，等］
Similarly,ibrtheoppositesides3and4andoppositesides5and6,wecanreadily
showthat．
〃…-`僻-`制し`…-`催-, 割
SummingaUthecontributionsgives
州一興[等一等ﾙ'謄一筈胖際一等］
(1.44）
SimUarprocedurescanbeusedtodctermineVxAinsphericalandcylindrical
coordinates・WemayalsolbrmthccrossproductoftheoperatorVwiththcvector
pointlimctionAbyemployingthecrossproductoftwovcctors・OnecaneasUy
showthatthisoperationgivesexactlyEq.(1.44)．
l８ｖＥｃｒｏＲＡＮＡＬＹｓＩｓ
…-賎l
Thcdeterminantmustbeexpandedmｔｅｒｍｓｏｆｔｈｅｔｏｐｒｏｗｔｏｈａｖｅｍｃａｎｉｎ9．
SinceVxAislbrmedlikeacrossproductoftwovectors,VandAitiscxpected
tohaveavalueindependentofthcparticularcoordinatesysteminwhichitis
representedWeshallsee，lbrexample，ｔｈａｔVxAhasaphysicalmeaningand
retainsthatmeaningindependentofｔｈｅsystemofcoordinatcsmwhichAiscx‐
pressed・Thus,weshallassumethatVxAisavector,justasweassumcdWwasa
OnecanlbrmaUyobtaindiHerentcoordinatereprcsentationsfbrVxAbywrit‐
ingtheoperatorVandthevectorAinaconsistentsctofcoordinates・Ｗｃｎｅedonly
partialderivativescqualtozcro(unlikcthccartesian{父,9,2}unitvectors,whosc
derivativesarealwayszero).Thismethodgivesthelbllowingexpressionlbrthecurl
”A-僻一等]+`[讐一劉化院１Ｍ′鶏](!"）
Equation(1.46)indicatesthatthercisnosimpledctcrminantnotationthatcanbe
appＵｅｄｔｏｔｈｅｃｕｒｌｏｆａｖｅｃｔｏｒｉｎｃｙｌｉｎｄｒicalcoordinates・However，ｏｎｅｃａｎｓｔｍ
…|； ;Ⅱ
Similarly，ｏｎｅｍａｙｄｅｒｉｖｅｔｈｅfbllowihg expressionfbrVxAinspherical
coordinates．
”A一命隈(い､昨念仏･)ﾙ`[志詩-号(M･)］
＋`隠仙汁淵
…満際l
(1.47）
SYoAes，Ｚ１ｂｅｏｒＥｍ・FinaUyweproveaveryusefi｣lrclationinmagnetostaticsthat
1.5ＴＨＥＣＵＲＬＡＮＤＳＴＯＫＥＳ，ＴＨＥＯＲＥＭ１９
＝、
ｅ
み､
ｎ
町”e1.IjDeterminationOfthecomponent
ofacurlofavectoralongaunitvcctoraby
appIyingthedefinitionofthecurltoadilL
lelcntialpillboxofnegligibleheight6,andaxis
alonge．
elementofsurfaccS,ｓｈｏｗｎｉｎＦｉｇ、114,ｉｎｔｈｅｉｂｒｍｏｆａｃｙｌｉｎｄｅｒｗｈｏｓｅｔｏｐａｎｄ
ｂottomeachofareaSbareperpendiculartoGandwhosesidesofwidth6andarea
Sbisparallelto6UsingEq.(1.43)lbrthisvolumeelement,ｗｅｈａｖｅ
▽…去曵…
ＷｅｎｏｗｃａｌｃｕｌａｔｅｔｈｅｃｏｍｐｏｎｅｎｔｏｆＶｘＡａｌｏｎｇ６､Ｔａｋｉｎｇｔｈｅｄｏｔｐｒｏｄｕｃｔｏｆ６
ｗｉｔｈＶｘＡｇｉｖｅｓ：
…A-鈍…△仏鯛）
｡……Ｗｌ戦岼~…T
If6issuflicientlysmall,ｔｈｅｎｏｎｅｗｒｉｔｅｓｄａ＝Jdlii,where‘lisanelementoflength
alongtheband,andiiisaunitvectornormaltotheband､Notingthatnx6＝i，
舵i縦:i蝿9搬雛雛競臘冊:蜜'…副…u…`nＡ
…A一三{",
ASD→0,scistheareaborderedbythepathofi､tcgrationandcontainsthevector
i・Thesenseofcirculationisrelatedtoeasthetumingofaright-handscrewistoits
advancaWrittcnlbrmally．
…-概臥A欧］川
whereSerepresentsanareawhosenormalisparaUeltoeandwhoseperimeterof
lengthCisthepathofintegrationNotethatthemtegraljcA･driscaUedthe
circulationofAarouｎｄＣＷｉｔｈＥｑ.(1.49)onecanreadilyobtainthccomponentsof
VxAinorthogonalcurviUnCarcoordinates．
Z0vEcToRANALYsIs
ＷｅｎｏｗｕｓｅＥｑ.(1.49)toprovcStokes，theorem,whichrelatcstheHuxofthecurl
ofAthroughasurfacetothecirculatioｎｏｆＡａｒｏｕｎｄｔｈｅｅｄｇｅｏｆｔｈｅｓｕｒｆａｃｅ・
ConsideranopensurlaceSwhoscperipheryisaclosedcurveC,ＷｅｔａｋｅＣｔｏｂｅ
ｓｊｍｐ/yco""ｃｃに｡;thatis,itcanbccontinuouslyshrunkdowntoapointwithoutthe
curvesleavmgthespacaThesurlaceShastwosides,oncofwhichwedeclarethe
positiveside,Wenowsubdividethissurlacemtovectorelcmentsofarea(ejAqj）
whichareessentiallyplanarifAａｊａｒｅｓｍａｌｌｅｎｏｕｇｈＦｏｒｅａｃｈｏftheseareaelements
wemayapplyEq.(149)ibrthecomponentofthecurlofAinthedirection6j;as
fO11ows：
ⅢA岼釧/W,。，mAM･’一妙Ｕ`O）
WenowlbrmthesumofthecxpressionｓｍＥｑ.(1.50)fbrqllthesurlaceelements
oftheｓｕｒｆａｃｅＳＴｈｕｓ
ｚＷＷ４･Fzf心`‘
ＩｎｇｏｉｎｇｔｏｔｈｅｌｉｍｉｔａｓＡａｊ→Ｏａｎｄｔｈｅｎｕｍｂｅｒｏｆｅｌｅｍｅｎｔｓｔｃｎｄｓｔｏinfinity,the
left-handsidebecomesanintegraloverthe(open)surfaceS､Theright-handside
bec･mesthelineintegralaroundcurveC,SincecontributionsoflA･`ribrallline
elcmentsinternaltoScancel,andonlythecontributionsoftheriｍｏｆＳ(thatis,on
qremain・Thcrelbre
LいAM-li
Ａ・aｒ
(1.51）
whichisknownasStokes，theorcmThepositivesideofSandthesenseinwhichC
istraversedarerelatedviatheright-handconvention．
AnimmediateutilizationofStokes，thcoremisthederivationofthecriterionfbr
determiningwhetherafieldisc…ｒｕＱ`iueornotJfiA．｡r＝0,fbrallpossible
closedpaths,inaregionofspace,thenitlbllowsthatVxA＝Ocvcrywhereinthis
region・Theconverseisalsotrue､Suchavectoriscalledaconservativcvector・We
maytherefbresummarizethecriteriathatdeterminewhetherornotavectorfieldA
isconservativeinaregionofspacelfinsomesimplyconnectedregiononeofthe
lbUowmgrelationsholds，
い-゜……
ＶｘＡ＝Ｏ
（1.52）
Ａ＝Wlbrsomescalarlimction/(seeExampleL2）
thenAisaconsewativefieldOneofthesccriteriabemgtruethroughoutasimply
connectedregionofspaceimpliｅｓｔｈａｔｔｈｅｏｔｈｅｒｔｗｏａｌｓｏａｒｅｔｒｕｅ．
L6VectorManipulationsofV
1.6.1SingIeDelOperations
Wehavediscussedthemeaningsofthegradient,divergcnce,andcurloperations､Ｉｎ
ｓｏｄｏｉｎｇｗｅｈａｖｅｃｏｍｅｔｏｒｅｇａｒｄｔｈｅdeloperator,Ｖ,asaquantitythatacquires
meamngonlybyoperatingonwhatis“totherightwofit,butotherwisebehavesin
1.6VECTORMANIPULATIONSOFV21
manylcspectslikeanordinaryvector・Ｗｅｗｉｓｈｈｃｒｅｔｏｓｕｍｍａｎｚｅｓｏｍｅｏｆｔｈｅｓe
operations・Ｉｆ/and9arescalarIimctions,andAandBarcvectorfimctionsof
positioninspace,theｎ
Ｖ(／＋9)＝Ｗ＋Ｖ,（1.53）
ｖ･(Ａ＋Ｂ)＝７Ａ＋ｖ･Ｂ
（1.54）
Ｖ×(Ａ＋Ｂ)＝ＶｘＡ＋ＶｘＢ
（155）
Ｖ(〃)＝(Ｗ)9＋/(V,）（1.56）
Ｖ･(/A)＝(Ｗ)．Ａ＋/(７Ａ）（157）
Ｖ×(/A)＝(Ｗ)ｘＡ＋八Ｖ×Ａ）（1.58）
Ｖ(Ａ･Ｂ)＝(Ｂ･Ｖ)Ａ＋Ｂｘ(ＶｘＡ)＋(Ａ･Ｖ)Ｂ＋Ａ×(ＶｘＢ）（1.59）
ｖ･(ＡｘＢ)＝(ｖｘＡ)･Ｂ－(ｖｘＢ)．Ａ
（1.60）
Ｖ×(ＡｘＢ)＝(Ｂ･Ｖ)Ａ＋Ａ(ＶＢ)－(Ａ･Ｖ)Ｂ－Ｂ(ワＡ）（1.61）
ThesefbrmulasmayallbeprovedbyexpressingVincartesiancomponentsand
comparingbothsidesofthcaboveequations．
1.6.2DoubleDelOperations
ThedeloperatormaybeappliedseveraltimesinsuccessionConsideringascalar
pointfUnction/ｗｅhave,fbrexample,Ｖ･WandVxWIncartesiancoordinates，
,w-(捌会刊÷@基)(A昊十'蟇柁妥）
SinceR･父＝，，＝２．２＝１and”＝父１２＝，.z＝0,ｗｃｈａｖｅ
ｌ
ｗ'一傷+暴劇ﾉｰ耐
(1.62）
ＴｈｉｓｉｓｃａｌｌｅｄｔｈｅＬａｐｌａｃｊα〃operator､TheLaplacianinothercoordinatescanbe
detcrminedusingsimilarprocedurcs・IncyUndricalandsphericalcoordinａｔｅｓｉｔｉｓ
ｇｌｖｅｎｂｙ
w-誌㈹+滞十裏
w一幕('２:O+論晶(…劉半論纂
(1.63）
(1.64）
ThecurlofthegradientofascalarVxWcanalsobedetermmedUsingsimilar
procedures・Incartesiancoordinatesone駒rites
２０｜ゐ可一生
，０｜〃郎一の
Ｗｗｌｌ
Z2vEcroRANALYsIs
Expandinggives
vx(w)-,偽-紛號，儂一紛十`偽-蒜）
butfbrwcll-bchaved,continuousfUnctions6ﾂ〕/Oy6z＝0伽zOy,etc.,thecurlofthe
gradientofascalarlimctionvanishes[seeEq.(1.52)definingconservativcvectors]；
thatｉｓ，
ＶｘＷ＝0（1.65）
Withavectorfieldfonecanlbrmthevariousdoubledclexpressions・Onecan
show,however,that(ＶｘＶ)･fan。(ＶｘＶ)xfvanishThequantityV･(Ｖｘｆ),the
divergenceofthccurlofavectorcaｎａｌｓｏｂｅｓｈｏｗｎｔｏｂｅｚｃｒｏｂｙｄｉｒｅｃｔｃａlculation
incartesiancoordmates；ｔｈａｔis，
Ｖ・Ｖｘｆ＝Ｏ
（1.66）
Equation(1.66)isimportantinmagnetostaticssincethedivergcnceofthemagnetic
fieldBisknowntobezero(Ｖ･Ｂ＝0),thenitallowscastingofthemagneticiieldB
intermsofavectorpotentialA：
Ｂ＝ＶｘＡ
FinallywediscussthecurlofthecurlofavectorVx(ＶｘＤ・Thisdoubledel
operationhaswideapplicationinthepropagationofelectromagneticwaves,atopic
tobediscussedinthelaterchaptersofthisbook・RegardingVasavectorlVx(Ｖ
ｘｆ）ｃａｎｂｅｅｘｐａｎｄｅｄｂｙｔｈｅｕｓｕａｌｔｒｉｐｌｅｖectorproductax(ｂｘｃ)＝b(a.c）
－(a.b)cThisgives
Ｖ×(Ｖｘｆ)＝Ｖ(Ｖ･f)－Ｖ.(Ｗ）
(1.67）
whereWisasecond-ranktensorordyadic(seeExampleL3).Incartesiancoordi‐
nateswehaveV．Ｍ)＝(Ｖ･Ｖ)f＝V2fwhereVzisthelaplacianoperator．
L7VectorlntegralRelations
HerewcdiscussanumberofextensionstothcdivergencetheoremandStokes，
theorcmAlthoughwewillnotneedalloftheseextensionsfbrthedevelopmentof
electricityandmagnetismatthelevelofthisbook,weincludethemfbrthesakeof
completenessandasamturerelbre､Ｃｅ．
DiDwgBDu“Ｚ１ＩｅＤ'Ｆ､、TheintegralrelationsgivenbelowinEqs.(1.68)ｔｏ(1.72)are
extensionsofthedivergencethｅｏｒｅｍ
ＬＭｗ川wM-l1owM‘（し`8）
ThisiscalledGreen，sfiriitidentityortheorem
L･閥,-ＷＭ－Ｌ(川-"｡Ⅳ。ｕ`'１
L７ｖＥｃＴｏＲＩＮＴＥＧＲＡＬＲＥＬＡＴＩｏＮｓ２３
ThisiscalledGreen,ssecondidentityorsymmetricaltheorem
llMv-炎(ｉＭ`･一生`…
作…L･iMo
1mwwM薑企EM1d。
(1.70）
(1.71）
(1.72）
Equations（1.68）ａｎｄ（169）canbeprovcdeasilybyapplyingthedivergcnce
theoremtothevcctorF＝⑪ＷａｎｄＦ＝⑩Ｗ'－しＶの,respcctivcly,wherCのand
しarescalarhmctions・
Equations（1.70）ａｎｄ（1.71）ｃａｎａｌｓｏｂｅｐｒｏｖｃｄｂｙａｐｐｌｙｉｎｇｔｈｅｄｉvergence
theoremtothevectorF＝ＡｘＣａｎｄＦ＝のＣ,respectivcly,whereCisaconstant
vector．
８ｍＡ“，Z1b2o肥JPu・ThelbllowingintegralrelationsarcextensionsofStokes，
theorem．
｛…-LＭ１…。
企…LいⅧ-1…
(1.73）
(1.74）
ＴｈｅｓｅｔｗｏｒｅｌａｔｉｏｎｓｃａｎｂｅｐｒｏｖｅｄｂｙapplyingStokes，thcoremtothevector
F＝ＢｘＣａｎｄＦ＝⑩Ｃ,respectively,whereCisaconstantvector．
ExampIeL1VeIocityFieldinaWaterDrain
A
Considcravectorfieldgivenbyv(r)＝poalBdshowninFig､1.15,whercpisthedistance
fromthezaxisのisaconstant,and巾isassociatcdwiththeangularcoordinatejaboUtthez
axislsvconservative？
い‘
－回
Ｒ(ﾊﾝｩ
殉”２１.JSVelocityfieldinawaterdrain．
2４vEcToRANALYsIs
Thatitisclcarlynotconscrvativcissccnltomconsideringitscirculationonacircularpath
ofradiusRaboutthezaxis：
ｊ……に."姉１Ｍ…，
Thecirculationisnonzero,ａｎｄthe”lbrevisnonconservative,Thisexamplemightbringto
mindthevelocityfieldofwatergoingdownthedraininasink・Thccirculationofthewateris
notingeneralzerolbrsuchasystcm．
Example1.2CCI聾ewativeNatureofRadiHIVector￣PotentialFunctions
ConsideraradialvectorfieldgiveninsphericalcoordinatesbyA＝／(r)fwhercノ(7)isa
scalarfimctionthatdependsonronly・Wewnlshowbelowthatthisvectorisconservative・
ThecriteriathatdeterminewhetherornotavectorisconservativearegiveninEq.,(152)．
SubstitutingA,＝/(7),andAo＝Ａの＝OinVxAinsphericalcoordinates(Eq､1.47)immedi‐
atelygivesVxA＝０;thusindicatingthatAisconservative
RadialvcctorIieldsarCofimportancetoelectmstaticssincetheelcctricfieldproduccdbya
pointchargeisradial,andhenceitisconservativeBecauseoftheimportanceofthisprop‐
erty,wewillexaminetheconservativenatureo｢thesevectorsfromthcpointo『viewofthc
lastcriterionofEq.(1.52).IfAisconscrvativc,thenitmustbewrittenasthegradientofa
scalarfimction⑩;thatis,Ａ＝Ｖ⑩､ThelimctionのiscalledapotentiaMimctioncorrespond‐
ｉｎｇｔｏＡ,Toshowthisweconsideralimction
｡-トーい`'
lftheintegralexistsandisacontinuoushmctionofr，thcnfromtheexpressionfbrthe
gradientinsphericalcoordinatesweｓｅｅｔｈａｔ
－－
０
一一
”｜姉
”｜”
７
１
１
一一
〈Ｊ
”一ケ
Ｔｈｅ艇sultisthatthereexistsahmctionのｓｕｃｈｔｈａｔ
Ｖ⑩＝ノ(r)f
(1.75)，
Aimdeedhasapotcntialfimction．
Example１．３GmdientofaVector-Dyadics
Thisexampledealswiththegradienｔｏｆａｖｅｃｔｏｒ,ｗｈｉｃｈｗｉｌｌｂｅｕｓｅｆＵｌｗｈｅｎｗｅｄｅａｌｗｉｔh
lbrcesonclectricdipolesplacedinextemalelectricfields・ConsideravcctoTE＝ＥｘＲ＋Ｅ,，
＋Ｅ爵2．Formally,ｗｅｃａｎｄｅｆｉｎｅＶＥａｓｆｂＩｌｏｗｓ：
Expanding,ｗｅｇＣｔ
,画一(m呈令，急糀呈)'……）
，膠一儂鰄+警船筈鋤)+傍，粁等，，半等'2）
＋(等鐡半警鰄÷料
(1.76）
Thequantities鰍,Ｒ,,…arccalledMPIitdWuｴﾉ３.NotcthatR,,lbrexampIe,isnotthesamc
aslf;thuswehavcnincdiflbrentunitdyadsinthegradient､Aquantitythatcanbeexpandediｎｔｈｅｌｂｒｍ
⑩＝a11jtR＋ｑ１ｚＲ,＋α1322＋α2,,2十α22”＋α2392＋α3,22＋α3229＋α33雌（1.77）
iscallCdadyadic,andtheninccocHicientsava”itscomponents．
1.8ＳＵＭＭＡＲＹ２５
ItisusclUltoexaminethescalarproductofavectorA＝Ａ』＋Ａｊ,，＋dz2andadyadic
のofthefbrmgivenabovc・ConsidertheproductA.⑩FormaUy,ｗｅwrite．
Ａ･⑩＝八雲１．⑩＋Ａ,,.⑩＋Ａｇａ．⑩
AsanexampIeconsidertheproductR･の.Theproducthasninetcrms,mcludingfbrexample，
父･q11jtA,父･q12fl,ｆ･口2,批,ａｎｄ父.α3ﾕ21.Theseindividualproductscanbeevaluatedusing
thefbllowingrules：
ｊｔ･q11RR＝α,,(2.2)Ａ＝α1,ｋ
ｋ.q12R,＝α12(R･父ルーｑ１２１
ｊｔ｡α2,,Ａ＝α2,(R･,)R＝０
２．α32”＝ﾛｺｺ(f･2),＝Ｏ
（1.78）
AnalogousrulesexistlbrtheICstoftheproductsandlbrtheproductsl･のａｎｄ２．⑩See
ProblemL20fbraspecificexample．
ExampleL4DiracDeltaFunction
InthisexamplewemtroduceaveryusefUlfimctionfbrdcalmgwithpointcharges・WewUl
dcnneithcrefromamathematicalpointofview･ItsrelevencetoelectmmagnetismwiUbe
introducedlatcr､TheDiracdeltafimction,giventhesymbolNr)isdefinedasfbllows：
刑＝o1brr≠0
0.79）
ル)`'-１（川
wherctheintegraliscarriedoutoverallspace､ThisdehnitionshowsthatthedeltalUnctionis
averyhighsingularmathematicallimction;itiszeroeverywhereexceptatasinglepointand
yethasanonzeromtegral("spike,，Iimction).ThisfUnctionisobviouslynotacontmuous
onc；thusitshouldnotbcdiHbrentiatcdasacontinuouslUnction・Neverthelessitisavery
usefUlmathematicalpropertyifhandledcautiousIy、
AnotherpropertyoftheDiracdeltafimctioncomesfromitsrelationtotheLaplacianor
divergenceoperator；ｔｈａｔｉｓ，
,声-,2e)-伽川
Ｉｔｉｓｅａｓｙｔｏｓｈｏｗｂｙｄｉ塵ctdilYerentiationthat
,声-vい÷響
SinceV．ｒ＝3,Ｖ(1/r3)＝－３０/P.4,thcnWvr3)iszeroIbrr≠Oandbecomesindeterminateas
r→OThenatureofthedivergCnceatr＝Ocanbcexaminedusingthedivergencetheorem・
ApplyingthetheoremtoasmallvolumeofradiusRgives
仁畠`｡一際`･一志}…
SincethisrcsultistrucregardlessofhowsmallR,thenonecanrCplacetheV･け3)by4元6(r)．
1.8Summary
Whenanelementofsurlacea琵adaatrisrcferredtotheorigin,itisoltenconvenienttouse
theconceptofsolidangledQ：
幻一等-…`,（,川24）
Z6vECToRANALYsIs
IfwehaveasurlaceSthatcompletelyenclosestheorigin,then｡＝fsdQ＝4祁,whereasifwc
haveaclosedsurlacethatdoesnotenclosetheorigin,thenＱ＝0．
ThegradientoperatorVisalineardiHerential``vectoroperator”thatcanoperateOn
scalaraswellasvectorfields/andAtogivethegradientofascalar,divergenceofavector，
curlofavectorbandgradientofavector．
ｗ一興二十，墓+魁筌
,A-筈十等十等
，…除一等ﾙ，謄一筈胖[警一等］
,A-謄鰄+等鋤十等鰯]+…
(1.30）
(1.41）
(1.44）
(1.76）
lnthelastequationtheellipsis(…)representsthreeanalogoustermsfbreachofthederiva‐
tiveswithresPccttoyandz・ThegradicntOperatormaybeapplieｄｓｃｖｃｒａｌｔｉｍｅｓｉｎｓｕｃ‐
cession,ormaybeappliedtoproductsoflimctions･Theoutcomeofsuchoperationscanbe
derivedfromtheabovebasicdilYbrentia］operations・OneimportantoperationistheLapJ
IacianoperatoractingonascalarorvectorlimctionVW＝ＷｏｒＶ２Ａ､Allopcrationscam
alsobederivedintermsofothercoordinatesystems(e9.,cylindricalandsphericalsystems)．
someintegralidentitiesofthedel(gradient)operationscanbederivedbymtegratingthe
dilYercntialonesoveranarbitraryvolumePboundedbyaclosedsurlaccS,ｏｒｏｖｅｒａｎｏｐｅｎ
ｓｕｒｆｎｃｅＳｂｏｕｎｄｅｄｂｙａclosedcurveCTheseincludethedivCrgencetheoremandStokes，
theorcm．
仁…尖…
[……い，
(1.42）
(1.51）
AvectorAissaidtobeconservativeiｆ
ＶｘＡ＝Ｏｏｒ
い-゜(．.……
(1.52）
Ifso,Ａｃａｎａｌｓｏｂｅｗｒｉｔｔｅｎａｓａｇｒａｄｉｅｎｔｏｆascalar
A＝一Ｖ⑪（conservativeＡ）
Problems
L1DeterminetheunitvectorperpendiculartotheplanethatcontainsthevectorsA＝Ｚｊｔ
－６１－３２ａｎｄＢ＝4Ａ＋３１－２．
１．２DetclmineanequationlbrtheplanepassingthroughthepointsP,(2,-1,1),Ｐ２(3,2
-1),ａｎｄＰ３(-1,3,2)．
l3ThcpositionvcctorsofpointsP1andP2areA＝３１t＋，＋22ａｎｄＢ＝jR-21-42
DetermineanequationfbrtheplanepassingthroughP2andperpendiculartotheline
joiningthepoints、
１．４（a)ShowthatVr風＝"r"~ｚｒ.(b)FmdVlnlrlandV(1/"、
nSConsiderthesurIacedefinedbythecquation2xzユー３ｘｙ－４ｘ－７＝OFindaunit
vectornormaltothissurfaceatthepoint(1,-1,2)．
１⑩Considerthelimction⑩＝xZyz3・InwhatdirectionliromthepointP(2,1,-1)isthe
directionaIderivativeof⑩ａｍａｘｉｍｕｍ？Whatisthemagnimdeofthismaximum？
句再。■伊閃筥唾い】
』勤
再酌
窪・三富ョ・｛号喫）Ⅱニマ洲○・
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騨Ｎ③『○ロー竜⑰『碗③己。⑪。
（輯）ロ鴛曾曰冒③弓③８罵言ロ露角一寺》自己⑤蒔馨③ご③具○再診Ⅱ（謀＋脚］＋疸些坤十｛胃’ぜ
１卿）》＋冠Ｈ＋こ＋鰐）》尉胃○一畳○目一一》・口器塁農ぐ⑨．ｅ）】）里③宮口旨③号○昏曰蔓ｏｐｅ》
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ニミ彦罠⑫芦。昌旦号③。。■駒冨昌色ず③萬号⑰急。（。『少Ⅱ（Ｈ＋凶』）浄十（』－凹卿）》十（罵十白蜘）騨雷
⑫○行曰。』□稗一（ず画⑫Ｎ③討○昌一く③吋媚③口。③）や
駒二．聿一曽貰号③く⑰⑤ご『画Ⅱ且『牌厨８口器弓自】く①．□③富】蔑昌回③ｅ碗巨＆昏筥閂Ⅱ－ぐｅ自已
ｅ（色）ⅡＰ望弓③昂ｐＶＣ・
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切豈○竜二皀弓③ご⑰。【。『抄Ⅱ（②韓苣＋町〕）沖十（］蔑凶１Ｎ）》十（］潴恥－竜）坤一⑫。○回⑫③『置菖ご⑰・田ご二
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号①８目⑰舌○口昌邑碩己Ｃ冨昌旨一言ロ三○国ｅ曽昌昏皀シⅡ『ｅ・
口③雷『ヨーロ③『・『曲已□岳③皆『雷。⑪一罠農『巴一助『・与冒・頭「意思唾蔚湧⑥一○麗○豊ユ画。③．
６○局一二曾号③ご⑱旦○再少Ⅱ』託鈴－吋、》＋闘念騨口旦忌③『③四○口け。■ご○⑰色ず嗜識』＋、Ⅱ一・
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軸Ⅱ。騨再○凶Ⅱ】。（聾）Ｏ巴⑪『ヨヨ③『・少．（ず）ロ⑰（。ｐ凰口⑪号⑰巨昌（『⑪○ｓ忌口○吋目囚］【○善③
皆ユ芭○囲い】（いⅡ。｝・駁（いⅡ』｝鱒ロー言⑥。胃ぐ⑰且豊熟画。。堕』｛ｘ凶十、Ⅱ」）．（ｏ）ぐ⑰ユ辱曽⑪
昌蔚『礪③旨⑰ｓ８忌己註『少冨百二○宮③『号③守○巨二②＆忌璽。■筈○ぐ②．
６。■⑫昼③『二③く③。ご『少Ⅱ墹誇十薦》－』宅』愚図ロ○号⑪⑫■里買溺。『（ず。○臣辱邑旦③『駒。⑰罰邑。』ず臣
批吋十託Ⅱ一○（回）ロ③（○閏昌再③二③巨已声ぐ③日。『ロ○回己騨三○一意⑫口且胃⑰。害意⑥里一言○国■陽画
冒再＆○二○『蔑騨且』．（ず）向く昌冨冨昌⑱旨１宵⑤冒冨関昌一“』，二畳。ご曾号⑰印三。⑥冨昌
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碗昌静。⑰。『号⑰昌一旨。③『ず③豆「⑪昌悶Ⅱ。酋己○町Ⅱ山．
６。■呂二曾二⑪ご③○ｓ『しⅡ（吋潴－］）声－》恥》－、璽夢”宮。、号③臣ごロ③『ずこ『駒■ユ四○③。『吾⑰
⑫宮：魏紬十、＋純いⅡ一．この『辱浬・穴⑪二言・『§・岸『ｘ夢・写冒Ⅱ｝、少・員壽：０－鰯
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号③ず。■■○餌昌○二二⑥豊『ず（稲吻・
勺『○ぐ①晉皀（⑬）一切②号Ⅱ。酋国己（す）｝『ｘ》』ロⅡ◎ず閂揮ロ竃⑤一○望邑旨塁質潴・
勺『○く③号昌一『ｘ再号Ⅱ鈩与ｘ因冒自皀（ご一『ｅ署Ⅱ一ｓ』』白・
ロ⑰［③員昌弓⑰『凶一口『》『凹型閏回已司凹｛」（『｝（『＃。．）
。○冨匡③『弓③昌舌＆○ｓⅡ浄蝉十二＋露（鷺浦田愚目ご］⑰】・】｝・口③【曾曰目⑰『．（ｅ・閂）閏口。
（『．ｅ）・閑・房二③忌画国電四目亘晦昌巨旨軍「閂三二媚『・ｅｐ『己
□輿⑪国ご旨⑰善⑰、園島③三○『『。

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