komnaekchblx '-ihcea, kbatephhohob cnhhob v.i. _ geometrija kompleksnyx chisel... · b. h....
TRANSCRIPT
![Page 1: KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB CnHHOB V.I. _ Geometrija kompleksnyx chisel... · B. H. APHOl\bA vn~~~~ KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB H CnHHOB H3j\ATEI\bCIllO MOCKOIICKOro](https://reader034.vdocuments.site/reader034/viewer/2022042318/5f071ec87e708231d41b67fb/html5/thumbnails/1.jpg)
B. H. APHOl\bA
vn~~~~
KOMnAEKCHblX'-IHCEA,
KBATEPHHOHOBH CnHHOB
H3j\ATEI\bCIllOMOCKOIICKOro ljEHTJl!l. HEnpEpblBHoroM"TEMATH~ECKOfOOOp.uoBAHHH
![Page 2: KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB CnHHOB V.I. _ Geometrija kompleksnyx chisel... · B. H. APHOl\bA vn~~~~ KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB H CnHHOB H3j\ATEI\bCIllO MOCKOIICKOro](https://reader034.vdocuments.site/reader034/viewer/2022042318/5f071ec87e708231d41b67fb/html5/thumbnails/2.jpg)
y II:K 514/515BBK 22.1
A84
KOMIIJIeKCHhIe 'fMCJIa OIIMChIBaIOT ,l1BM:lKeHMJI eBKJIM,l10BOM IIJIOCKOCTM, 0,l1
HOMy Bpaw;eHMIO TpexMepHoro IIpOCTpaHCTBa COOTBeTcTByeT zma rena-rep
HMOHa, pa3JIM'fMe KOTOphIX (cPM3MKM Ha3BaJIM 3TO JIBJIeHMe CIIMHOM) CBJI3aHO
co CBOMCTBaMM rpYIIIIhI IIpe06pa30BaHMM. "BpaII\eHMJI» 3JIeKTpOHOB OTJIM'fa
IOTCJI OT BpaII\eHMM TBep,l1hIX TeJI MMeHHO pa3JIM'fMeM CIIMHOB, MrpaIOII\MX pe
IIIaIOII\YIO POJIb IIpM OIIMcaHMM 3JIeKTpOHHhIX 060JIO'feK aTOMOB.
B 6poiliIOpe, napa.ay C OCHOBHhIMM cPaKTaMM KJIaCCM'feCKOM TeopMM KOM
IIJIeKCHhIX 'fMCeJI M KBaTepHMOHOB, paCCKa3aHhI nexor-opsre HOBhIe peayas
TaThI M rMIIOTe3hI. HaIIpMMep, KOMIIJIeKCHOM aepcaea .re-rpaaapa OKa3hIBa
eTCJI OKTa3,l1p, a rMIIOTe3a, 'fTO KBaTepHMOHHaJI er-o BepCMJI - MKOCa3,l1p,
He ,l10Ka3aHa.
TeKCT 6POiliIOPhI IIpe,l1CTaBJIJIeT C060M ,l10IIOJIHeHHYIO 06pa60TKY 3aIIMCM
JIeKIIMM, IIpO'fMTaHHOM B. H. ApHOJIb,l10M ,l1JIJI IIIKOJIbHMKOB 9-11 KJIaCCOB
17 HOJI6pJI 2002 roaa na MaJIOM MeXMaTe J\1TY.
Bpouuopa pacc-rar-aaa na IIIMPOKMM xpyr- 'fMTaTeJIeM, MHTepecyIOII\MXCJI
MaTeMaTMKoM: IIIKOJIbHMKOB CTapIIIMX KJIaCCOB, cTY,l1eHTOB, y'fMTeJIeM ...
ISBN 5-94057-025-9
Apno.abO B.aaou,M,up Heopeeusc
reOMeTpMJI KOMIIJIeKCHhIX 'fMCeJI, KBaTepHMOHOB M CIIMHOB
Pe,l1aKTOp B. B. Tpywno6. Texa. penaxrop B. M. Fupoeuu,3aIIMcb M pacmarpponxa JIeKIIMM: P. 11. Bozdano«, M. P. Boeoonoe.
JIMIIeH3MJI H~ N'01335 OT 24/Ill 2000 r-,
Cl>opMaT 60 x 88 1/16. Cl>M3. rre-r . .II. 2,50.
IIo,l1IIMcaHO K IIe'faTM 5/Ill 2002 r-,
YCJI. rre-r. .II. 2,50. Tapaac 1500 3K3.
H3,l1aTeJIbCTBO J\10CKoBcKoro IIeHTpa nenpepsmnoro MaTeMaTM'feCKOrO 06pa30BaHMJI.
121002, Mocxaa, r-2, BOJI. BJIaCbeBCKMM nep., 11.
OTIIe'faTaHO B TMIIorpacPMM P A "Cl>aHTa3MJI».
111024, Mocxaa, E-24, ABMaMOTOpHaJI yJI., 20/17.
![Page 3: KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB CnHHOB V.I. _ Geometrija kompleksnyx chisel... · B. H. APHOl\bA vn~~~~ KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB H CnHHOB H3j\ATEI\bCIllO MOCKOIICKOro](https://reader034.vdocuments.site/reader034/viewer/2022042318/5f071ec87e708231d41b67fb/html5/thumbnails/3.jpg)
Teopaa KOMIIJIeKCHhIX 'IYlCM, KBaTepHYlOHOB Yl CIIYlHOB OTHOCJlTCJI K He
60JIbIIlOMY 'IYlCJIy HaYl60JIee ¢YH,L\aMeHTaJIbHhIX 'IaCTeM reOMeTpYlYl, YlMeIO
II:\YlX HaYl60JIee Ba2KHbIe IIpYlJI02KeHYlJl B ¢Yl3YlKe. OIIYlCbIBaeMOe HYl2Ke r-eoxeeTpYl'IeCKOe IIOCTpOeHYle TeOpYlYl KOMIIJIeKCHbIX 'IYlCM ¢paHIJ;Y3bI IIPYlIIYlCaJIYl
Apr-any, XOTJI aa CeMb JIeT ,L\O Apr-ana ero oIIy6JIYlKOBaJI ,L\aTCKYlM lVIaTelVIa
TYlK BeCCMb (KOTOpbIM, BIIpO'IeM, YlCXO,L\YlJI cxopee Yl3 6Y,L\yII:\Ylx YlH2KeHepHhIX
IIpYlJI02KeHYlM, narrpmeep, K eII:\e He IIOCTpoeHHOM TOr,L\a TeOpYlYl YPaBHeHYlM
MaKCBMJIa Yl rrepexeemror-o TOKa).
KOMILJIeKCHhIe ~I'ICJIa
PaCCMOTpYlM na eBKJIYl,L\OBOM IIJIOCKOCTYl CYlCTeMY OpTOHOpMYlpOBaHHbIX
KOOp,L\YlHaT.
1
PMC. 1. BeII\eCTBeHHhIe 6a3McHhIe nex-ropsr na IIJIOCKOCTM KOMIIJIeKCHhIX 'fMCeJI.
Ba3YlCHbIe BeKTOpbI 0603Ha'IYlM -repea 1 IIO O,L\HOM OCYl, i IIO ,L\pyroM OCYl
(OT CJIOBa «imaginary», T. e. MHYlMbIM). TO'IKa IIJIOCKOCTYl IIpe,L\CTaBJIJleTCJI
B BYl,L\e a + b . i (e,L\YlHYlIJ;y IIpYl a He IIYlIIleM):
b
1 a
PMC. 2. BeII\eCTBeHHaJI M MHMMaJI -rac-rr, KOMIIJIeKCHOro 'fMCJIa.
Beicropsr na IIJIOCKOCTYl M02KHO CKJIa,L\bIBaTb:
1*
al + b: . ia2 + b2 · i
3
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KpOMe CJI02KeHYlJl KOMlIJIeKCHhIX 'IYlCM, OlIpe,LIMYlM YlX YMH02KeHYle.
Ta6JIYlIJ;a YMH02KeHYlJl:
1·1 1, 1· i t . 1.
CalVIOe rJIaBHOe Ha'IYlHaeTCJI npa YMH02KeHYlYl i na i:
i·i -1.
L[YlCJIO i Ha3hIBaIOT M'/iUMbIM, TaK KaK He CYII:\eCTByeT BeII:\eCTBeHHoro 'IYl
CJIa a, ,LIJIJI xor-opor-o BhIlIOJIHJlJIOCb 6hI paBeHCTBO
IIpoYl3Be,LIeHYle ,LIBYX JII06hIX KOMlIJIeKCHhIX 'IYlCM OlIpe,LI6JIJleTCJI lIO 3aKOHy
,LIYlCTpYl6yTYlBHOCTYl:
HHhIMYl CJIOBaMYl, YMH02KaJI al na a2 Yl b1 na b2 , lIOJIy'IaeM BeII:\eCTBeHHYIO
'IaCTb lIpOYl3Be,LIeHYlJl (YlX pa3HOCTb), a YMH02KaJI al na b2 Yl b1 na a2
MHYlMYIO 'IaCTb (CYMMY ClTYlX ,LIBYX lIpOYl3Be,LIeHYlM). OlIpe,LIMeHYle YMH02KeHYlJl
3aKOH'IeHO.
3aMe~aHl'le. Bce CBOMCTBa YMH02KeHYlJl (KOMMYTaTYlBHOCTb: ZlZ2
Z2Z1; aCCOIJ;YlaTYlBHOCTb: (ZlZ2)ZS Zl(Z2ZS); ,LIYlCTpYl6YTYlBHOCTb lIO OT-
HOIIleHYlIO K CJI02KeHYlIO: Zl (Z2 + ZS) ZlZ2 + ZlZS) BhIlIOJIHJlIOTCJI O'IeBYl,LIHhIM
06pa30M.
C aJIre6paYl'IeCKOM TO'IKYl apenas TeOpYlJl KOMlIJIeKCHhIX 'IYlCM ClTYlM YlC
'IeplIhIBaeTCJI.
,.lJ;BI'I~eHI'IH ILJIOCKOCTI'I
KOMlIJIeKCHhIe 'IYlCJIa - MaTeMaTYl'IeCKYlM armapar ,LIJIJI OlIYlCaHYlJl ,LIBYl
2KeHYlM lIJIOCKOCTYl. L[To6hI B ClTOM y6e,LIYlTbCJI, MhI BBe,LIeM eme ,LIOlIOHYl
TMbHoe
Orrpezrezremre. KOMlIJIeKCHOe 'IYlCJIO
Z a-b·i
Ha3hIBaeTCJI '1WMnJl,e",;C'/iO conpJ!;)ICe'/i'/ibIM K 'IYlCJIy Z a + b . i.
4
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PMC. 3. KOMIIJIeKCHOe COIIpJDKeHMe KOIIIKM K.
reOMeTpM~eCKM rrepexo~ OT Z K Z -- 3TO OTproKeHMe OTHOCMTeITbHO
OCM Ol.Teopexra. ConpJI/JlCe'li'liOe "iUCJW ",; cy.M.Me d6YX "';O.M,nJl,e"';C'libIX wuce« pa6
no cy.M.Me wucea conpJ!;)/Ce'li'libIX ",; CJI,aZae.M,bl.M,:
Teopexra. ConpJ!;)/Ce'li'lioe "iUCJI,O ",; npouseedewuso d6YX "';O.M,nJl,e"';C'libIXwucea paeno npouseedewuso wuce« conpJ!;)/Ce'li'libIX ",; .M,'lio;)/CumeJl,Jl,.M,:
Orrpeztezremre. IIpoM3Be~eHMe KOMrrJIeKCHOrO ~MCJIa na COrrpJl2KeHHOe
eMy ~MCJIO Ha3bIBaeTCJI "';6adpamO.M, .M,OdYJl,J! KOMrrJIeKCHOrO ~MCJIa:
JIeMMa. M OdYJl,b "';O.M,nJl,e",;c'liozo "iUCJI,a -- deUCm6UmeJl,b'liOe 'lieompuv,ameJl,b'liOe "iUCJI,O.
,ZJ;oKa3aTe.JIhCTBO. Kaanpar MO~YJIJI He MeHJleTCJI npa COrrpJl2KeHMM:
ZZ ZZ ZZ,
noa'rouy OH JlBJIJleTCJI ~eMCTBMTeITbHbIM ~MCJIOM. KpOMe Toro, IzI 2ZZ
a2 + b2 ~ 0, noa'rouy MO~YJIb -- T02Ke ~eMCTBMTeITbHOeHeOTpMU;aTeITbHOe
~MCJIO. DOrrpezrezremre. Apzy.M,e'lim a He paaaoro HyJIIO KOMrrJIeKCHOrO ~MCJIa
pasea yrJIy noaopo'ra OT rrOJI02KMTeITbHOM noayoca 01 B CTOpOHy rrOJI02KM
TeITbHOM MHMMOM nozyocx Oi ~o HarrpaBJIeHMJI KOMrrJIeKCHOrO ~MCJIa.
5
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3aMe~aHHe. ECJIM Izl 1,TO
a cos o, b sin o.
Te, KTO He 3HaKOM C ¢YHKIJ;MJlMM CMHYC M KOCMHyC, MoryT C'IMTaTb ClTO
3aMe'IaHMe MX onpe,LI6JIeHMeM.
IIpMMeHMM Tenepb KOMnJIeKCHbIe 'IMCJIa ,LIJIJI M3y'IeHMJI ,LIBM2KeHMM eBKJIM
,LIOBOM nJIOCKOCTM. PaCCMOTpMM KOMnJIeKCHOe 'IMCJIO w E C. PaCCMOTpMM
npeocpaaoaaaxe «YMH02KeHMe na KOMnJIeKCHOe 'IMCJIO z», nepeao.aamee xa-
2K,LIyIO TO'IKy W B TO'IKy ZW, r,LIe Izi 1, argz a.
Teopexra. Ilpeotipasoeawue YMnO;)ICenu,J! na '1WMnJ/,e'KCnoe "iUCJ/,O C MOdYJ/,eM edunu'/{a ,J!6J/,J!emc,J! n060pomOM neocxoctmi {w}.
,ZJ;oKa3aTeJIhCTBO. Paccsrorpmc KOMnJIeKCHOe 'IMCJIO w. COC'IMTaeM
MO,LIYJIb KOMnJIeKCHOrO 'IMCJIa, B xoropoe nepexonar W npa HarrIeM npeo6pa
30BaHMM:
ZWZW (zz) . (ww) WW
CJIe,LIOBaT6JIbHO, JII060M aercrop nepexomrr B sercrop TaKOM 2Ke ,LIJIMHbI. KpoMe Toro, paCCTOJlHMe Me2K,LIy KOHIJ;aMM BeKTopOB TaK2Ke COXpaHJleTCJI:
PMC. 4. OilepaIIMJI YMHO:lKeHMJI na KOMIIJIeKCHOe 'fMCJIO C MO,l1YJIeM e,l1MHMIIa.
TaKMM 06pa30M, npeotipaaoaaaae YMH02KeHMJI na KOMnJIeKCHOe 'IMCJIO
C MO,LIyJIeM e,LIMHMIJ;a COXpaHJleT ,LIJIMHbI.
Ba2KHaJI ,LIeTaJIb: ClTO npeotipaaoaaaae COXpaHJleT OpMeHTa!J;MIO. D3a~a~a. BpaII:\eHMe nJIOCKOCTM { W } no 'IaCOBOM CTp6JIKe nepexomer npa
HarneM npe06pa30BaHMM BO BpaII:\eHMe no 'IaCOBOM CTp6JIKe (T. e. B Ty 2Ke
CTOpOHy, 'ITO M MCXO,LIHoe).
OTcTyILJIeHHe npo opHeHTa:U;HIO. II:JIJI onpe,LI6JIeHMJI opMeHTaIJ;MM
Hy2KHa ¢opMyJIa, KOTOpyIO OT cTy,LIeHTOB 3a'IacTyIO CKpbIBaIOT, - ¢opMyJIa
6
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rrJIOII:\a,lV'l napaaaenor-pasoea. Ilycr-s na eBKJIYl,L\OBOM rrJIOCKOCTYl C OpTOHOp
MYlPOBaHHbIMYl KOOp,L\YlHaTaMYl {(X, Y)} YlMeeTCJI napaaaenorpaaoa. IIepBbIM
BeKTOp, 3a,L\aIOII:\YlM napanaezor-pasoa, 0603Ha'IYlM -repea A (Xl, yd, BTO
POM - -repea B (X2,Y2).
Y
X
PMC. 5. 0pMeHTal\MJI rrJIOCKOCTM napoir BeKTopOB.
Teopexra. Ilnouuui« S(A, B) napa.n,.n,eofWZpa.M.Ma, nopoocdewuoeo 6e'Kmopa.MU A U B, ,j/,6.f1,,}!emc,j/, .n,u'lieu'liou ¢Y'Ii'K'U,ueu om eexmopa A :
~~~~~~~~~~~:/
((xl,yd = A
PMC. 6. IIapaJIJIeJIorpaMM, rrOpO:lK,l1eHHhIM napoir BeKTopOB A M B.
IIJIoII:\a,L\b Ha,L\O C'IYlTaTb co 3HaKOM (<rrJIIOC», eCJIYl noaopo'r OT A K B B HarrpaBJIeHYlYl BpaII:\eHYlJl OT rrepBoM KOOp,L\YlHaTHOM rrOJIyocYl KO BTOPOM
(aa Ha:uIeM pacymce - (<rrPOTYlB 'IaCOBOM CTPeJIKYl»). CooTBeTcTBeHHo,
co 3HaKOM «MYlHYC», eCJIYl noaopo'r OT A K B - B rrpOTYlBOrrOJI02KHYIO CTO
pony. JIYlHeMHOCTb 3aBYlCYlMOCTYl rrJIOII:\a,L\Yl OT nepaor-o sercropa oaaaxae'r
TaK2Ke, 'ITO
S(kA,B) kS(A, B).
3TYl ,L\Ba rrpOCTbIX ¢aKTa CO,L\ep2KaT B CKpbITOM BYl,L\e BCIO «TeOpYlIO onpene
JIYlTeJIeM» .
7
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B03bMeM 6a:3l'IC e (1,0), f (0,1). Tor-na HarrIM aercropsr A AlM B A2 M02KHO rrpe,LICTaBMTb B BM,LIe
BbI'IMCJUleM rrJIOIIJ;a,LIb S(A I , A2 ) . BCJIe,LICTBMe JIMHeMHOcTM rrOJIy'IaeM CYMMY
'IeTbIpeX CJIaraeMbIX
3,L1eCb S(e,e) 0, no-touy 'ITO napaazezor-pasoe, HaTJlHYTbIM na rrapy (e,e),JlBJIJleTCJI BbIp02K,LIeHHbIM. ITo nOM 2Ke caMoM npa-mae S(f, f) o. lI:aJIee
3aMeTMM, 'ITO S (e, f) 1; HO S (f, e) -1, TaK KaK nanpaaneaae BpameHMJI
OT f K e - B o6paTHyIO CTOpOHy. CJIe,LIOBaTeJIbHO, BCJI rrJIOIIJ;a,LIb panna
dTO 'IMCJIO Ha3bIBaeTCJI onpedeJl,UmeJl,eM rrpMBe,LIeHHOM HM2Ke KBa,LIpaTHOM
Ta6JIMIJ;bI M3 'IeTbIpeX KOMrrOHeHT HarrIMX aercropoa, Ha3bIBaeMOM Mampuv,eunapaaaenor-pasoea:
I:~ ~~ I·MbI o6cy2K,LIaeM sonpoc: COXpaHJleTCJI JIM npa YMH02KeHMM na z OpMeHTaIJ;MJI?
Ha,LIo B3J1Tb na rrJIOCKOCTM {w} OCHOBHOM napaanezor-paxor M rrOCMOTpeTb,
Ky,LIa OH rrepeXO,LIMT. ECJIM rrJIOIIJ;a,LIb oopaaa rrOJI02KMTeJIbHa, TO 0pMeHTaIJ;MJI
COXpaHJleTCJI, a eCJIM rrJIOIIJ;a,LIb oopaaa OTpMIJ;aTeJIbHa, TO 0pMeHTaIJ;MJI OCHOB
HOrO (a 3Ha'IMT, M JII06oro) napaaneaorpacoea MeHJleTCJI. 06Pa3bI BeKTopOB
e w ] CYTb
ze z a + bi, zf zi -b + ai.
YI)Y2
Orrpe,LIeJIMTeJIb :'lTOM MaTpMIJ;bI rrOJI02KMTeJIeH (npa z ::j:. 0), TaK KaK
Ia b I a2
- ( -b) . b a2 + b2 ~ o.-b a
8
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CJIe,LIOBaTMbHO, YMH02KeHMe na HeHyJIeBOe KOMrrJIeKCHOe 'IMCJIO Z coxpa
HJleT OpMeHTaU;MIO rrJIOCKOCTM YMH02KaeMbIX 'IMCM.
I1:cCJIe,LIyeM, na KaKOM yrOJI rrOBOpa'IMBaeTCJI BeKTOp W npa ero yMH02Ke
HMM na z? B03bMeM rrpOCTeMIIlYlM BeKTOp M rrOC'IMTaeM, na KaKOM yrOJI OH
rrOBepHeTCJI npa namese rrpe06pa30BaHMM.
3aMe~aHl'le.AJIre6paMCTbI C'IMTaIOT rrpOCTeMIIIMM 'IMCJIOM 'IMCJIO 0, HO
HaM OHO He rrO,LIXO,LIMT, MbI B03bMeM B Ka'IeCTBe npocreamer-o 'IMCJIO w 1.Tor-na YMH02KeHMe na z nepesene'r ero B zw z a + bi.
PMC. 7. Iloaopor nercropa W IIpM YMHO:lKeHMM na 'fMCJIO z,
11:3 aror-o CJIe,LIyeT, 'ITO HaIIIe npeoopaaoaaaae rrOBOpa'IMBaeT BeKTOp
w 1 na yr-oa a argz.
CJIe~CTBl'le.YoMiliO;)ICe'liUe na '1WMnJ/,e'KC'lioe 'liUCJ/,O z, ma'Koe 'limo Izl 1,,j/,6J/,,j/,emc,j/, n060pomOM neocxoctm: na ueo« pa6'1iblU arg z.
Teopexra. IIpu YM'lio;)ICe'liuu 'KOMnJ/,e'KC'libIX wucea ux apzYMe'limbl C'KJ/,aoueosomc»:
arg(zw) argz + argw.
,ZJ;OKa3aTeJIhCTBO. MbI Y2Ke,LIOKa3aJIM, 'ITO BeKTOp w 1 npa yMH02Ke
HMM na z rrOBOpa'IMBaeTCJI na yroa a. Ho rrOCKOJIbKY YMH02KeHMe na z eCTb
noaopor, TO na TaKOM 2KeyrOJI rrOBOpa'IMBaeTCJI M JII060M BeKTOp w. BBe,LIeM
0603Ha'IeHMe arg w (3. Tor-na 'IMCJIO zw MMeeT apryMeHToM a + (3, T. e.
arg(zw) argw+a. DdTO rrpMBO,LIMT K TpMrOHOMeTpM'IeCKMM T02K,LIeCTBaM, xo'ropsre MHa'Ie
HeB03M02KHO noasra. Ilycr-s z a + bi, w e + di, Izl 1, Iwl 1. Tor-na
zw (ae - bd) + (ad + be)i.
IIOCKOJIbKY arg z a, TO a COS a, b sirr a. TO'IHO TaK 2Ke, rrOCKOJIbKY
arg w (3, TO e COS (3, d sirr (3. IIwToMy rrOJIy'IaeM ,LIJIJI BeIIJ;eCTBeHHbIX
M MHMMbIX 'IaCTeM rrpOM3Be,LIeHMJI zw BbIpa2KeHMJI
2-KOMII.'f.
cos(a + (3)
sirr(a + (3)
ae - bd
ad+ be
COS a cos (3 - sirr a sirr (3,
cos a sirr (3 + sirr a cos (3.
9
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<f:>opMyJIa YMH02KeHYlJl KOMrrJIeKCHhIX 'IYlCM JIerKO 3aIIOMYlHaeTCJI Yl He rry
TaeTCJI HYl C 'IeM ,L\pyrYlM. A TpYlrOHOMeTpYl'IeCKYlX ¢OpMyJI B HeM conep2KYlTCJI O'IeHb MHoro, 'ITO ,L\MaeT Yl3 HYlX Yl3JII06JIeHHbIM ClK3aMeHaTOpaMYl BO
rrpoc na npaeuasrx ClK3aMeHax.
Teopaa BpaII:\eHYlM Yl ,L\BYl2KeHYlM eBKJIYl,L\OBOM rrJIOCKOCTYl YlC'IeprrbIBaeTCJI
rrpYlBe,L\eHHbIMYl ¢OPMYJIaMYl.
3a~a~a. 1I:0Ka2KYlTe, 'ITO npu .fI/1060M 'liamypa.n.b'liOM n
cos(n<p) + i sirr(n<p) (cos <p + i sirr <p)n
(ara ¢opMyJIa Ha3bIBaeTCJI (/JOpMy.n.OU Mya6pa, TaK KaK ee OTKpbIJI ropaano
paasme COBceM ,L\pyroM 'IMOBeK).
IIpHMep. 113 ClTOM ¢OPMYJIbI CJIe,L\yeT (rryTeM pacxparras cK060K B 6YlHOMe npaaoii 'IaCTYl), 'ITO Yl cos (n<p) , Yl sirr(n<p) - BeII:\eCTBeHHbIe lVIHOrO
'IJIeHbI C IJ;MbIMYl KOCl¢¢YlIJ;YleHTaMYl OT rrepexcemnax x COS ip, Y sirr ip,
HJIYl, YlHbIMYl CJIOBaMYl, mpueonouempuuecwue M'liOZO"i.n.e'lib! (JIYlHeMHbIe KOM
6YlHaIJ;YlYl CYlHyCOB Yl KOCYlHyCOB KpaTHhIX yrJIOB) MO;)IC'liO paccuasnpuecm» .,,;a.,,;
Ozpa'liU"ie'liuJ! 06b!"i'lib!X M'liOZO"i.n.e'li06 om d6YX nepeuenstua: (x U y) na 0"';-
pY;)IC'/iOCmb (x2 + y2 1).
B 'IaCTHOCTYl, O'IeHb 'IaCTO rrOJIe3HbI ¢OPMYJIbI
cos(2<p)
cos(3<p)
cos2 <p - sirr' ip ;
4 cos" <p - 3 cos ip,
sirr(2<p)
sirr(3<p)
2 sirr <p cos ip,
3 sirr <p - 4 sirr3 ip,
IIOCKOJIbKY KOCYlHyC - 'IeTHaJI ¢YHKIJ;YlJl, a CYlHYC - He'IeTHaJI ¢YHKIJ;YlJl
OT ip, TO cos (n<p) M02KHO rrpe,L\CTaBYlTb B BYl,L\e lVIHOrO'IJIeHa OT O,L\HOrO JIYlIIlb
apr-yxrem-a x cos <p (3aMeHYlB Be3,L\e sirr' <p na 1- cos2 <p). 3TYl 3aMe'IaTMb
HbIe lVIHOrO'IJIeHbI OT O,L\HOM nepexceanoii Ha3bIBaIOTCJI lVIHOrO'IJIeHaMYl L[e6bI
meaa, OHYl 06JIa,L\aIOT MHOrYlMYl rrOJIe3HbIMYl CBOMCTBaMYl (<HaMMeHee yKJIOHJI
IOTCJI OT HyJIJl», ,L\OCTaBJIJlIOT «¢YlrypbI JIYlCCa2KY» na axpaae OCIJ;YlJIJIorpa¢a*
Yl T. ,L\.). IIpOCTeMIIlYle Yl3 ClTYlX lVIHOrO'IJIeHOB nepenacsraaiorcs B ,L\pyrYlx 060
3Ha'IeHYlJlX CJIe,L\YIOII:\YlM 06pa30M:
4x3 - 3x,
O,L\HaKO, C casroro Ha'IaJIa B03HYlK sonpoc: KpOMe BpaII:\eHYlM Yl ,L\BYl2KeHYlM
rrJIOCKOCTYl JR2 CYII:\eCTBYIOT BpaII:\eHYlJl Yl ,L\BYl2KeHYlJl rrpOCTpaHCTBa JR3. KaK
* CM., narrpastep, B. N. Ap1i.O.f1,bO, J\1aTeMaTM'feCKMe MeTO,l1hI KJIaCCM'feCKOM MexaHMKM(M.: 3,l1MTOpMaJI YPCC, 2000).
10
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MX OrrMChIBaTb? KOe-'ITO 3,L1eCb C,LI6JIa.JI yace BeCC6JIb rrpmeepno B 1820 ro,LIY.
Ho OKOH'IaT6JIbHyIO TeopMIO rrOCTpOMJI rr03,L1Hee MpJIaH,LICKMM MaTelVIaTMK
raMMJIbToH.
O,LIHMM TyMaHHhIM Be'IepOM no rrYTM M3 TpMHMTM-KOJIJIe,LILKa B lI:y6JIMHe
raMMJIbTOH rrpnfier- K rrOMOII:\M a.JIKOrOJIbHhIX napoa. dTO M rrpMB6JIO ero K aa
Me'IaT6JIbHOM TeopeMe, KOTOpyIO MhI M3JIOLKMM HMLKe (M KOTOpyIO OH rrhITa.JICJI
nepen ClTMM HaMTM MHOrO JIeT). rOBOpJlT, 'ITO OH 6hIJI TaK nopaacen OTKphI
TOM MM ¢OPMYJIOM, 'ITO CeM'IaC LKe BhIpe3a.JI ee rrep0'IMHHhIM HOLKMKOM na ne
paaax ,LIepeBJlHHOrO MOCTMKa -repea KaHa.JI, no KOTOpOMy OH B ClTO BpeMJI
II1eJI. Ho 51, XOTb M MCKa.JI CITY Ha,LIrrMCb, BhIpe3aHHYIO TOr,LIa raMMJIbTOHOM,
He HarnM ee na CITOM MOCTMKe*.
0606m;eHl'le KOMILJIeKCHhIX ~I'ICeJI ~o IIOHHTI'IH KBaTepHI'IOHOB
OKa3hIBaeTCJI, ,LIJIJI OrrMCaHMJI BpaII:\eHMM B JR3 HyLKHhI 4 'IMCJIa, rrOClTOMY
HaM Tpe6yIOTcJI aercropa 4-MepHoro BeII:\eCTBeHHOrO rrpOCTpaHCTBa - 'K6amepwuouu. BhIPe3aHHaJI raMMJIbTOHOM na nepaaax ¢opMyJIa ijk -l,L1aeT
Ta6JIMIJ;y yMHoLKeHMJI KBaTepHMoHoB.
HTaK, KBaTepHMoH - ClTO aercrop 4-MepHoro BeII:\eCTBeHHoro npocrpaa
CTBa C 6a3MCOM 1, i, j, k (xo-ropsae Ha3hIBaIOTCJI 6a3MCHhIMM KBaTepHMo
HaMM): a + bi + cj + dk. L[MCJIO a Ha3hIBaeTCJI eeuiecmeennotl "iacmb'lO(C'Ka.r!JI,PO.M,) , a TpexMepHhIM aercrop v bi + cj + dk - .M,1iU.M,OU "iacmb'lOKBaTepHMoHa. CJIOBO «aex'ropt srrepasre rrOJlBMJIOCb MMeHHO B CITOM TeopMM.
Bo apexrena raMMJIbTOHa BeKTopOB He 6hIJIO, TaK 'ITO eMy rrpMII1JIOCb BhI,LIy
MhIBaTb BCIO TepMMHOJIOrMIO.
«L[MCJIa» 1, i, j, k Ha3hIBaIOTCJI 6a3UC1ibl.M,U 'K6amep1iU01ia.M,u.CJIOLKeHMe ,LIBYX KBaTepHMoHoB - rrOKOMrrOHeHTHoe. OHO KOMMyTa
TMBHO M aCCOIJ;MaTMBHO.
Caxcoe Tpy,LIHOe - rrpM,LIyMaTb, KaK yMHoLKaTb KBaTepHMoHhI. YMHoLKe
HMe ,LIMCTpM6yTMBHO OTHOCMT6JIbHO CJIOLKeHMJI, TaK 'ITO ,LIOCTaTO'IHO yMeTb
yMHoLKaTb 6a3MCHhIe KBaTepHMOHhI.
* OrIMcaHHhIM 3IIM30,l1 IIpOM30IIIeJI 16 OKTJI6pJI 1843 r-ona na Bpyxr-aucxosi MOCTy -re
pes KopoJIeBcKMM KaHaJI, r,l1e raMMJIbTOH M BhIpe3aJI cPOpMyJIy ~jk -1. B IIMCbMe
caoesty ChIHY Ap'fM6aJIb,l1y OT 5 anr-ycra 1865 r-ona raMMJIbTOH IIMIIIeT 0 cY,l1b6e
3ToM cPOpMyJIhI: «... Ho, KOHe'fHO, KaK Ha,l1IIMCb, OHa y ace CTepJIaCb». (JI. C. IIo.Ilan, BapMaIIMoHHhIe IIpMHIIMIIhI MexaHMKM, MX pa3BMTMe M IIpMMeHeHMe B cPM3MKe,
M.: Cl>M3MaTJIMT, 1960, 103-104.) - IIpuoM. pea.
2* 11
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Ta6JIYlIJ;a raMYlJIbTOHa YMH02KeHYlJl 6a3YlCHbIX KBaTepHYlOHOB:
1 ·1 1, 1 . i i, 1 . j j, l·k k·,i . 1 i, i . i -1, i . j k, i . k -j;
j ·1 j, j . i -k, j.j -1, j·k i:,k ·1 k, k· i j, k·j -i, k·k -1.
3aMeTYlM, 'ITO paaasre MHYlMbIe KBaTepHYlOHHbIe «e,LIYlHYlIJ;bI» npa yMH02Ke
HYlYl He KOMMyTYlpyIOT, a aHTYlKOMMYTYlPYIOT: ZJ k, HO ji -k. ECJIYl
3HaTb, 'ITO ij k, TO OCTaJIbHOe JIerKO BbIBeCTYl Yl3 yCJIOBYlJl aCCOIJ;YlaTYlBHO
CTYl YMH02KeHYlJl. Hanpmrep, ik iij -j, rrOCKOJIbKY i 2 -1.IIpaBYlJIo YMH02KeHYlJl 6a3YlCHbIX KBaTepHYlOHOB rrOJIy'IaeTCJI Yl3 ¢OPMYJIbI
ij k IJ;YlKJIYl'IeCKYlMYl nepecraaoaxaae:
ij k, jk i, ki j.
YMH02KeHYle He KOMMyTaTYlBHO, HO aCCOIJ;YlaTYlBHO. HeKOMMYTaTYlBHbIM
OHO Yl ,LIOJI2KHO 6bITb, 'ITo6bI onacsraart, BpaII:\eHYlJl TpeXMepHoro npocrpaa
CTBa, xo'ropsre He ace KOMMyTYlpyIOT.
II:JIJI JII06bIX KBaTepHYlOHOB p, q, r YlMeIOT MeCTO CJIe,LIYIOII:\Yle paBeHCTBa:
p+q
(p + q) + r
(p. q) . r
p' (q + r)
q + P (KOMMyTaTYlBHOCTb no CJI02KeHYlIO),
p + (q + r) ( aCCOIJ;YlaTYlBHOCTb no CJI02KeHYlIO),
p . (q . r) (aCCOIJ;YlaTYlBHOCTb no YMH02KeHYlIO),
p . q + P . r (,LIYlCTpYl6YTYlBHOCTb).
TaKaJI onepanas YMH02KeHYlJl He Tpe6yeT YMCTBeHHbIX cnocooaoc-rea,
a rrO,LI06Ha YMH02KeHYlIO MHOr03Ha'IHbIX 'IYlC6JI*.
Ilycr-s ,LIaMbI ,LIBa KBaTepHYlOHa,
IIoc'IYlTaeM YlX npoaaaeneane. Ha'iHeM C BbI'IYlCJIeHYlJl BeII:\eCTBeHHOM 'IaCTYl
OTBeTa:
Re(pq)
* L\eKapT rrpe,l1JIO:lKMJI MeTO,l1 rrOJIHOrO MCKJIIO'feHMJI reOMeTpMM M Boo6pa:lKeHMJI M3 Ma
TeMaTMKM, KaK caMhIM «,l1eMOKpaTM'feCKMM»: nem. JII060M TyrrMIIa rrpO,l1BMraeTCJI rrpa
rr0,l106HOM MeTO,l1e CTOJIb »ce ycrrenmo, KaK M caMhIM reHMaJIbHhIM yM.
12
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JICHO, 'ITO npoaaaeneaae BeII:\eCTBeHHhIX 'IaCTeM ala2 B OTBeT BOM,LIeT, HO npa
rrepeMH02KeHYlYl 'IJIeHOB C ,LIpyrYlMYl 6a3YlCHhIMYl KBaTepHYlOHaMYl B03HYlKaIOT
i2 -1, P -1, P -1, a ,LIpyrYlx BeII:\eCTBeHHhIX CJIaraeMhIX B rrpo
Yl3Be,LIeHYlYl HeT. ITo onpenezemno, C'IWJI,,j/,P'lib!,M, npouseeoewuesc ,LIBYX BeKTO
pOB TpeXMepHoro eBKJIYl,LIOBa rrpOCTpaHCTBa C OpTOHOpMYlpOBaHHhIM 6a3YlCOM
(i, j, k) Ha3hIBaeTCJI 6YlJIYlHeMHaJI ¢YHKIJ;YlJl OT ClTYlX BeKTopOB
3a~a~a. lI:oKa3aTb, 'ITO CKaJIJlpHOe nponaaeneaae ,LIBYX BeKTopOB paa
HO rrpOYl3Be,LIeHYlIO YlX ,LIJIYlH na KOCYlHyC yrora Me2K,LIy HYlMYl:
BhI'IYlCJIYlM MHYlMYIO 'IaCTb rrpOYl3Be,LIeHYlJl KBaTepHYlOHOB pq.
Im(pq)
r,LIe [Vl,V2] - aercropaoe npoaaaenenae BeKTopOB Vl Yl V2. KOIvIIIOHeHThI
BeKTopHoro rrpOYl3Be,LIeHYlJl JIerKO BhI'IYlCJIYlTb - ClTO orrpe,LIeJIYlTeJIYl:
3a~a~a. lI:oKa3aTb, 'ITO [Vl, V2] nepneanaxyaspao Vl Yl V2.
BhI60p narrpaaaenas nepneanaxynapa rrO,LI'IYlHeH Tpe60BaHYlJlM opaea
TaIJ;YlYl (<<rrpaBYlJIO npaaoii pyrca»}.
TpOMKa BeKTopOB (Vl, V2, [Vl, V2]) opnem-apyer rrpOCTpaHCTBO TaK 2Ke,
KaK 6a3YlCHaJI TpOMKa (i, j, k). dTO 3Ha'IYlT, 'ITO O,LIHY Yl3 TpoeK M02KHO nenpe
phIBHO nepernny-rt, B ,LIpyryIO, OCTaBJIJIJI aexropsr JIYlHeMHO He3aBYlCYlMhIMYl
BO Bce BpeMJI neperar-aaaaaa. Hanpmrep, TPOMKYl (i,j,k) Yl (j,k,i) opaea
TYlPYIOT rrpOCTpaHCTBO O,LIYlHaKOBO, a (i,k,j) - naaxe.
Bexropnoe npoaaaenenae ,LIBYX BeKTopOB MeHJleT 3HaK npa YlX nepecra-
HOBKe.
IIpHMep. [i,j] k -[j,i], [i,i] O.ITpOYl3Be,LIeHYle JIYlHeMHO no Vl Yl no V2.
dTO onncaaae YMH02KeHYlJl aaaepmaer ormcaaae aJIre6phI KBaTepHYlOHOB
no raMYlJIbTOHY. OCHOBHoe anaxenae ClTOM onepamor COCTOYlT B ,LIOCTaBJIJle
MOM eIO onacaaaa BpaIIJ;eHYlM TpeXMepHoro eBKJIYl,LIOBa rrpOCTpaHCTBa. Harro
MHIO, 'ITO apamenae eBKJIYl,LIOBOM OpYleHTYlpOBaHHOM rrJIOCKOCTYl JR2 na yrOJI a
13
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3a,L\aeTCJI OT02K,L\eCTBJIeHYleM rrJIOCKOCTYl JR2 C C Yl npeoopaaoaanaen yMHO-
2KeHYlJl na Z, W f---+ ZW, r,L\e Z cos a + i sirr a.
k
v = (cosa)i + (cos{3)j + (cos"'()k
PMC. 9. OpT OCM BpaII\eHMJI M er-o HarrpaBJIJIIOII\Me KOCMHyChI.
II:JIJI KBaTepHYlOHOB M02KHO npenyr-anart, KBaTepHYlOHHbIM aHaJIOr 'IYl
CJIa z: Ho 3aKaH'IYlBaIOII:\ee BbI60p KBaTepHYlOHa 3aMe'IaHYle rrpYlBO,L\YlT K Te
opaa emma (BepOJlTHO, rrepBOOTKpbIBaTeJIeM ¢OPMYJIbI ,L\JIJI ner-o JlBJIJleTCJI
Ponpnr-ec}.MbI Ha'IYlHaeM C eBKJIYl,L\OBa OpYleHTYlpOBaHHOrO rrpOCTpaHCTBa JR3 C op
TOHOpMYlpOBaHHbIM 0pYleHTYlpOBaHHbIM 6a3YlCOM i, j, k. Bpamenae rrpo
CTpaHCTBa JR3 onpenenaercs CBoeM OCbIO BpaII:\eHYlJl Yl yrJIOM noaopo'ra BO
xpyr- :'lTOM OCYl. OCb M02KHO 3a,L\aTb OpTOM v, KOTOpbIM 3a,L\aeTCJI CBOYlMYl
yrJIaMYl a, {3, "'( C OCJlMYl (i, j, k): v i COS a + j COS {3 + k COS "'(. Bpameaaena rrOJI02KYlTeJIbHbIM yrOJI aoxpyr- ClTOM OCYl coananaer C BpaII:\eHYleM na rrpo
TYlBOrrOJI02KHbIM yrOJI aoxpyr- rrpOTYlBOrrOJI02KHOrO nanpaaaeaas OCYl.
PaCCMOTpYlM OpT OCYl BpaII:\eHYlJl. dTOT BeKTOp YlMeeT KOMrrOHeHTaMYl
KOCYlHyCbI HarrpaBJIJlIOII:\YlX yrJIOB (o6pa30BaHHbIX BeKTopOM Yl HarrpaBJIJlIO
II:\YlMYl BeKTOpaMYl oceM). IIo TeopeMe Ilarpar-opa, ,L\JIYlHa aercropa v panna
e,L\YlHYlIJ;e.
IIepeXO,L\YlM K penraromea ¢opMyJIe -,L\JIJI KBaTepHYlOHa, OrrYlCbIBaIOII:\erO
apameaae. dTy ¢OpMyJIy CKpbIBaIOT OT MexrvIaTJlH YlJIYl ¢Yl3YlKOB (B TeOpYlYl
TBep,L\OrO TeJIa paCCKa3bIBaIOT B BYl,L\e rvIaTpYlIJ; IIaYJIYl).
Kaxona pa3MepHOCTb r-pyrmsr acex BpaII:\eHYlM eBKJIYl,L\OBa rrpOCTpaHCTBa
JR3? MbI yno-rpeonaeu TpYl HarrpaBJIJlIOII:\YlX KOCYlHyca Yl yroa noaopo'ra, IIo
JIy'IaeTCJI 'IeTbIpe BeII:\eCTBeHHbIX 'IYlCJIa. Ka2KeTCJI, 'ITO pa3MepHOCTb MHO
r-ooopaana acex BpaII:\eHYlM panna 'IeTbIpeM, HO ClTO He BepHO. ECTb COOTHO
meaae na 3TYl 4 'IYlCJIa: Ilvll 1. Pa3MepHOCTb r-pynnsr 80(3) BpaII:\eHYlM
eBKJIYl,L\OBa rrpOCTpaHCTBa JR3 aoxpyr TO'IKYl 0 panna TpeM.
3a~a~a. IIOC'IYlTaTb pa3MepHOCTb rpyrrrr 80(n) acex BpaII:\eHYlM rrpo
CTpaHCTB JR4, JR5 Yl T.,L\. aoxpyr Ha'IaJIa KOOp,L\YlHaT.
14
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B 0603Ha'IeHMM SO(n) 6yKBa 0 0603Ha'IaeT OpTOrOHa.JIbHOCTb, T. e. CO
XpaHeHMe rrpe06pa30BaHMHMM rpyrrrrM ~AMH BeKTOpOB M3 ~n, a 6yKBa S(<crreU;Ma.JIbHoe») - COXpaHeHMe OpMeHTaIJ;MM.
Tenept, MM COrrOCTaBMM Ka2K~OMY BpaII:\eHMIO TpeXMepHoro opaea'rapo
BaHHOrO eBKAM~OBa rrpOCTpaHCTBa KBaTepHMOH HOpMM e~MHMu;a.
OIIpe~e.JIeHHe. Conp,j/,;)ICe'li'lib!M K KBaTepHMoHy q a + v Ha3MBaeTCH
KBaTepHMOH q a-v.OIIpe~e.JIeHHe.Keaopamosc 'li0PMb! KBaTepHMOHa q Ha3MBaeTCH 'IMCAO
q. q a2 - (v, (-V)) a2+ IIvI1 2 ~ o.dTO - acer-~a BeII:\eCTBeHHoe 'IMCAO, OHO rrOA02KMTMbHO npa q ::j:. O. lI:eM
CTBMTMbHO, HarrMCaHHOe BMpa2KeHMe eCTb CYlVIlvIa KBa~paTOB BeII:\eCTBeHHMX
'IMCM
II03TOMY M nopsra - BeII:\eCTBeHHoe neorpauaremsaoe 'IMCAO.
IIYCTb ~aH KaKOM-AM60 KBaTepHMOH C HOpMOM e~MHMu;a,
q a + v: Ilqll 1.
IIo onpeneaeamo HOpMM
ECTeCTBeHHO npaaan, a M Ilvll aa KOCMHyC M CMHyC nexo'ropor-o yrvra:
a cos o, Ilvll sin e.
MM rrpe~CTaBMAMHam KBaTepHMOH B BM~e q cos <p +sirr <p'Vi, r ~e Vi - OpT
OCM. BeKTOp TpeXMepHoro rrpOCTpaHCTBa Vi MMeeT HOpMy e~MHMu;a. ECAM
MM XOTMM OrrMCMBaTb npa rrOMOII:\M namero KBaTepHMOHa BpaII:\eHMe Tpex
MepHoro rrpOCTpaHCTBa, TO B Ka'IeCTBe op'ra Vi eCTeCTBeHHO B3HTb OpT OCM
BpaII:\eHMH. II:AH no.ny-reaas KBaTepHMOHa OCTa.JIOCb BM6paTb B rrOCAe~HeM
¢opMyAe yro.a ip, IIYCTb (} - casra BMM'IMHa yrora noaopo'ra, COBpeMeHHoe
rrOHMMaHMe ¢M3MKM OTAM'IaeTCH OT rrOHMMaHMH ¢M3MKM ~Ba CTOAeTMH aaaan,
noa'rouy Hy2KHa «~BOMKa Ponpar-eca»: OKa3MBaeTCH, B Ka'IeCTBe <p ~AH no
CTpoeHMH KBaTepHMOHa, orrMCMBaIOII:\erO noaopo'r na yr-oa (} aoxpyr OCM Vi,
Hy2KHO BM6paTb nozoaany yrora noaopo'ra, <p (}/2,
q cos (~) + sirr (~) . Vi.
II:BoMKa Ponpar-eca M eCTb npa-rana emma B ¢M3MKe.
15
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3aMe~aHl'le. H3-3a ,L\BOMKYl BpaIIJ;eHYle Orrpe,L\6JUIeT KBaTepHYlOH HeO,LI
H03Ha'iHO. YrOJI noaopo'ra 8 onpeneaea no MO,L\YJIIO 2nn, n E Z. BXO,L\J1IIJ;YlM
B orrpeneneaae KBaTepHYlOHa yrOJI 8/2 orrpenenen no MO,L\YJIIO ttn, ECJIYl n He
'leTHO, TO npa rrpYl6aBJIeHYlYl 2nn K yrJIy 8 KBaTepHYlOH Yl3MeHYlT 3HaK. Hano6hLJIO 6hI nacar-s ±q cos(8/2) + sirr(8/2) . Vi.
HHhIMYl CJIOBaMYl, O,L\HOMY Yl TOMy .lKe noaopo-ry OTBe'laIOT ,L\Ba xaarep
HYlOHa (pa3JIYl'laIOII:\YleCJI 3HaKoM).
Bce KBaTepHYlOHhI HOpMhI e,L\YlHYlU;a (11qll 1) 06pa3yIOT crpepy 8 3 B JR4.
CooTBeTCTBYle, COrrOCTaBJIJlIOIIJ;ee Ka.lK,L\OMy KBaTepHYlOHy HOpMhI e,L\YlHYlU;a
BpaIIJ;eHYle, 3a,L\aeT o rofipazaeaae crpepsr 8 3 na BCIO rpyrrrry BpaIIJ;eHYlM
,L\BYJIYlCTHOe naxpsrrae: 83 ---+ 80(3) 83 / ± 1.Teopexra. Omotipaucenue 'lia'Kpb!mu,J! ,J!6Jl,,J!emc,J! ZOMOMOp(/JU3MOM, m. e.
npouseedewue nepezodurn 6 npouseedenue: ecsni g(q) - 6paw,e'liUe, coomeemcmetnouiee 'K6amep'liuo'liy q, mo
YMHO.lKeHYle KBaTepHYlOHOB J1BJIJleTCJI Mre6paM'leCKOM 3aIIYlCbIO yMHO
.lKeHYlJl BpaIIJ;eHYlM (ara rpyrrrra HeKoMMyTaTYlBHa). dTO aHMOr TeopeMhI
CJIO.lKeHYlJl apr-yteerrroa npa rrepeMHO.lKeHYlYl KOMrrJIeKCHhIX 'lYlCM. Jlertco ,L\O
Ka3hIBaeTCJI CJIe,L\YIOIIJ;aJI
JIeMMa·lq1q21 Iq11·lq21·IIwToMy, eCJIYl Iq11 1, TO YMHO.lKeHYle na TaKOM KBaTepHYlOH COXpaHJleT
,L\JIYlHhI. OTCIO,L\a JIerKO BhIBeCTYl ¢OPMYJIy (*) (51 nponymy B ClTOM BhIBO,L\e
nexo'ropsre ,L\eTMYl).
Ilycr-s Z - KBaTepHYlOH Yl Izi 1. BY,L\eM ,L\eMCTBOBaTb KBaTepHYlOHOM
z na KBaTepHYlOH W CTpaHHhIM 06pa30M:
Orrpeztezremre. 06pam'lib!M 'K6amep'liUO'liOM K KBaTepHYlOHy Z Ha3hIBa
eTCJI TaKOM KBaTepHYlOH Z-l, 'ITO Z-l Z 1.TaKOM 06paTHhIM KBaTepHYlOH e,L\YlHCTBeHeH Yl ero JIerKO HaMTYl ,L\JIJI JIIO
60ro HeHyJIeBOrO KBaTepHYlOHa Z TaKYlM .lKe 06pa30M, KaK ClTO ,L\MaeTCJI
Yl ,L\JIJI KOMrrJIeKCHhIX 'lYlCM (<YlHBepCYlJl»). B HauIeM cay-rae npocro Z -1 Z,
TaK KaK ZZ 1. CTpaHHOe npeotipaaoaaaae (**) - no BpaIIJ;eHYlJl rrpo
CTpaHCTBa JR4 (,L\OKa3aTMbCTBO aHMOrYl'lHO CJIy'laIO CC).ECJIYl W 1, TO npa ClTOM BpaIIJ;eHYlYl 1 f---+ Z . 1 . Z, TO eCTb aercrop 1
oCTaeTCJI npa npeofipaaoaaaaa (**) na MeCTe. BpaIIJ;eHYle eBKJIYl,L\OBa rrpo
CTpaHCTBa JR4 OCTaBJIJleT na MeCTe e,L\YlHYlu;y. Opror-oaansnoe ,L\OrrOJIHeHYle
16
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K aercropy 1 nepexomrr B ce6H M BpaIIJ;eHMe rrpOCTpaHCTBa JR4 3a,LIaeT BpaIIJ;e
HMe eBKJIM,LIOBa rrpOCTpaHCTBa JR3. M02KHO HerrOCpe,LICTBeHHo npoaepara, 'ITO
TaK rrOJIy'IaeTCH TO CalVIOe BpaIIJ;eHMe TpeXMepHOrO rrpOCTpaHCTBa, KOTOpOe
MbI paasme OrrMCbIBaJIM npa rrOMOIIJ;M OCM M yrora noaopo-ra,PaCCMOTpMM ,LIeMCTBMe (**) ,LIJIH rrpOM3Be,LIeHMH ZlZ2: ZlZ2W(ZlZ2)-1.
KaK BbI'IMCJIMTb (ZlZ2)-1?IIo,LIcKa3Ka M3 3aJIa: (ab)-l a-1b-1.
- Heaepno!IIpHMep. Ilyc-rs a oaaaxaer CHHTb rrM,LI2KaK, a b - CHHTb pytiamxy.
Tor-na ba - CHHTb rrM,LI2KaK, 3aTeM CHHTb py6aIIIKy - ClKBMBaJIeHTHO pa3
,LIeTbCH. Ocparnoe - O,LIeTbCH. Ha,LIo CHa'IaJIa Ha,LIeTb py6aIIIKy, M JIMIIlb
rrOTOM Ha,LIeTb rrM,LI2KaK. IIoClToMY (ba)-l a-1b-1.
CJIe,LIOBaTeJIbHO, rrpOM3Be,LIeHMe ZlZ2 rrepeBO,LIMT TO'IKy W B TO'IKy
Onepanna (**) ,LIeMCTBMH KBaTepHMOHa ZlZ2 ClKBMBaJIeHTHa TOMy, 'ITO na pe
ayJIbTaT onepamor (**) ,LIeMCTBMH KBaTepHMOHa Z2 na W ,LIeMcTByeT onepa
U;MeM (**) KBaTepHMOH Zl:
HHbIMM CJIOBaMM, ,LIJIH .mofior-o 'IMCTO MHMMOrO (opTOrOHaJIbHOrO K 1) BeK
Topa W MMeeT MeCTO T02K,LIeCTBO
(9(Zl)) [(9 (Z2)) (w)].
B 60JIee KOpOTKMX 0603Ha'IeHMHX, BpaIIJ;eHMH 9, cooTBeTcTByIOIIJ;Me KBa
TepHMoHaM Z HOpMbI e,LIMHMU;a, y,LIOBJIeTBOpHIOT yCJIOBMIO rOMoMoP<PHOCTM,
TaK 'ITO Tpy,LIHO OrrMCbIBaeMOe YMH02KeHMe BpaIIJ;eHMM 9 (z) CBe,LIeHO K JIerKO
BbIrrOJIHMMoMY rrepeMH02KeHMIO cooTBeTcTByIOIIJ;MX MM KBaTepHMoHoB.
dTOT KBaTepHMoHHbIM MeTO,LI OrrMcaHMH ,LIBM2KeHMM MCrrOJIb3yeTcH ,LIa2Ke
B KOCMM'IeCKMX MCCJIe,LIOBaHMHX, npa opraHM3aU;MM opMeHTaU;MM cnyrnaxoa.
lI:oKa2KeM 06eIIJ;aHHOe BbIIIle COBrra,LIeHMe onepamor (**) ,LIeMCTBMH KBa
TepHMoHa
Z cos (8f2) +v'sirr(8f2) , Iv'l 1,
3-KOMII. 'f. 17
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C rrOBOpOTOM na yrOJI () aoxpyr OCYl ero MHYlMOM 'IaCTYl (c OpTOM Vi) B rrpo
CTpaHCTBe JR3 'IYlCTO MHYlMhIX KBaTepHYlOHoB.
1) OCb opma Vi nepexonar npa :'lTOM onepamor B ce61I. lI:eMCTBYlTMbHO,
YlMeIOT MeCTO CJIe,L\YIOIIJ;Yle TpYl O'ieBYl,L\HhIe ¢aKTa.
a) YMHoLKeHYle na BeIIJ;eCTBeHHoe 'IYlCJIO cos(()/2) nepeacmrr B ce61I JIIO
6yIO rrpoxO,L\JlIIJ;YIO -repea HOJIb npasryro.6) YMHoLKeHYle na 'IYlCTO MHYlMhIM KBaTepHYlOH Vi nepeecmrr KBaTepHYl
OHhI OCYl C OpTOM Vi B KBaTepHYlOHhI C HyJIeBOM MHYlMOM 'IaCTbIO (rrOCKOJIbKy
aercropaoe npoaaaeneaae JII06oro aercropa na ce6J1 paBHO HyJIIO).
B) 06paTHhIM K Z KBaTepHYlOH Z-l Z YlMeeT TaKOM LKe, KaK Z, BYl,L\
(c rrpOTYlBOrrOJIOLKHhIM ()), rrOClTOMY YMHOLKeHYle na Z-l cnpaaa 06JIa,L\aeT,
KaK Yl YMHOLKeHYle na Z CJIeBa, CBOMCTBaMYl a) Yl 6).
YTBepLK,L\eHYle 1) HerrOCpe,L\CTBeHHo BhITeKaeT Yl3 ¢aKToB a), 6) Yl B).
2) L[To6hI ,L\OKa3aTb, 'ITO yrOJI noaopo'ra BpaIIJ;eHYlJl (**) aoxpyr OCYl
op'ra Vi pasea (), ,L\OCTaTO'IHO rrpYllvleHYlTb CITy onepamno K KaKoMy-JIYl60 'IYl
CTO MHYlMOMY KBaTepHYlOHy W Yl3 opTOrOHaJIbHOrO ,L\OrrOJIHeHYlJl K OpTy Vi
B TpeXMepHOM eBKJIYl,L\OBOM rrpOCTpaHCTBe JR3 'IYlCTO MHYlMhIX KBaTepHYlO
HOB.
YMHOLKeHYle W na Z CJIeBa noaopa-oraaer ClTOT BeKTOp BHYTpYl yxaaaa
HOrO opTOrOHaJIbHOrO ,L\OrrOJIHeHYlJl na yrOJI () /2 (cOrJIaCHO TeopeMe 0 CJIO
LKeHYlYl apr-yverrroa npa nepesorozseaaa KOMrrJIeKCHhIX 'IYlCM). YMHoLKeHYle
na Z-l cnpaaa - TOLKe noaopo'r na () /2 (aanpaxcep, ClTO MOLKHO BhIBeCTYl Yl3
Toro, 'ITO
WZ ZW,
a 'IYlCTO MHYlMhIe KBaTepHYlOHhI npa COrrpJlLKeHYlYl MeHJlIOT 3HaK). YTBepLK,L\e
nae 2) ,L\OKa3aHO.
3aMe~aHHe. B KBaHToBoM ¢Yl3YlKe (nanpauep, npa onacaaaa BpaIIJ;e
HYlJl ClJIeKTpoHoB) OKa3hIBaeTCJI, 'ITO ¢Yl3Yl'IeCKYl BaLKHhIM JlBJIJleTCJI He ClJIe
MeHT r-pynnsr BpaIIJ;eHYlM 80(3), a YlMeHHO O,L\YlH Yl3 ,L\BYX COOTBeTcTByIO
IIJ;YlX eMy KBaTepHYlOHOB HOpMhI e,L\YlHYlU;a, KOTOphIM Yl Ha3hIBaeTCJI cmmos«3Jl,e",;mp0'lia, YlMeIOIIJ;YlM ,L\Ba 3Ha'ieHYlJl (06hI'IHO 0603Ha'IaeMhIe B ¢Yl3YlKe -re
pm ±1/2).IIpYl OTOLK,L\eCTBJIeHYlYl B O,L\HY TO'IKy KaLK,L\OM napsr rrpOTYlBOrrOJIOLKHhIX
TO'IeK crpepsr S" B eBKJIYl,L\OBOM rrpOCTpaHCTBe JRn+l Yl3 crpepsr rrozry-ra
eTCJI n-MepHoe rJIa,L\KOe MHOroo6pa3Yle, xo'ropoe Ha3hIBaeTCJI 6eUfeCm6e'li'lib!,M, n-,M,ep'lib!,M, npOe",;mU6'1ib!,M, npocmpcncmeos; Yl 0603Ha'iaeTCJI -repea JRpn(pac. 9).
18
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CKJIeHBaeMbleTO'lKH ±A
l\eJ)(PHHHM rrJIOCKOCTb JR2
(i6eCKOHe'lHO-y,n;MeHHM»rrpaxraa JRp1
0 1
B I
oopaa OKpeCTHOCTH3KBaTOpa
(rrOC.IIe CKJIeHBaHHH)
PMC. 9. Ilocr-poeaae rrpoercr-aanoa rrJIOCKOCTM rrpMKJIeMBaHMeM JIeHThI Me6Myea K xpyry AISIC I (MJIM rrpoercr-aaaoa rrpJIMOM lR!.P 1 «6eCKOHe'fHO- y ,l1a.rreHHhIX»
TO'feK K acPcPMHHOM rrJIOCKOCTM lR!.2) .
dTO MHOroo6pa3He M02KHO TaK2Ke onacam, KaK MHOroo6pa3He BCeX rrpo
XO,n;HIIIHX -repea 0 npnxnax OM B 06'beMJIIOIIIeM rrpOCTpaHCTBe JRn+1: Be,n;b
TaKM npmraa KaK paa H onpeneaaercs napoii ±N CBOHX (rrpOTHBOrrOJI02K
HbIX) TO'leK nepecexeaas C e,n;HHH'lHOM C¢epOM.
IIpHMep. IIpoeKTHBHM npsxeaa JRp1 eCTb OKPY2KHOCTb 8 1, rrOCKOJIbKY
npa OT02K,n;eCTBJIeHHH TO'leK <p H <p+ tt OKpy2KHOCTH {<p mod 21f} noay-raercs
OKPY2KHOCTb {<p mod 1f} 8 1/ ±1 ~ 81.
IIpoeKTHBHM rrJIOCKOCTb JRp2 noay-raercs H3 a<P¢HHHOM rrJIOCKOCTH JR2,n;o6aBJIeHHeM (i6eCKOHe'lHo-y,n;MeHHOM rrpHMOM» JRp1, conepzaameii no od'liou6eCKOHe'lHo-y,n;MeHHOM TO'lKe na Ka2K,n;OM rrpHMOM a<P¢HHHOM rrJIOCKOCTH
(,n;BHrMCb B,n;OJIb rrpHMOM B o6e CTOpOHbI, MbI rrpH,n;eM B O,n;HY H Ty 2Ke
6eCKOHe'lHO-y,n;MeHHyIO TO'lKY).
L[To6bI Bce 3TO HCHO yBH,n;eTb, M02KHO Ha'laTb, OT02K,n;eCTBJIHH npo'rnao
rrOJI02KHbIe TO'lKH He co BceM crpepsr 8 2 , a JIHilib C 3aMKHyToM nonycrpepr,r 8 2
(CKa2KeM, I02KHee 3KBaTopa). Tor-na CKJIeHBaTb rrpH,n;eTCH TOJIbKO Ka2K,n;yIO
TO'lKy A 3KBaTopa C rrpoTHBOrrOJI02KHOM, -A, a OTKpbITM crpor-o I02KHM
noayopepa npa CKJIeHBaHHH He nocrpanaer (pHC. 9).Mezsny npo-noa, H3 3TOM 2Ke KOHCTpyKIlHH BH,n;HO, 'ITO OKpeCTHOCTb
6eCKOHe'lHo-y,n;MeHHOM (a 3Ha'lHT, H JII060M) rrpHMOM na rrpoeKTHBHOM rrJIOC
KOCTH JRp2 ,n;H¢¢eOMop¢Ha JIeHTe Merinyca (KOTOPYIO Metiayc TaK H OT
KPbIJI) , BC.lIe,n;CTBHe 'lerO rrpoeKTHBHM rrJIOCKOCTb JRp2 neopaem-apyeua
(KaK H ace 'leTHOMepHbIe npoeicraamae rrpOCTpaHCTBa JRp2n H B OTJIH'lHe
3* 19
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OT He'IeTHOMepHhIX JRp2n+l, KOTOphIe BCe opaem-apyexna).IlTaK, 80(3) 83/±1 JRp3 - opaem-apyeuoe TpexMepHoe npoex
TYlBHOe rrpOCTpaHCTBO. Ero M02KHO rrpe,LICTaBYlTb ce6e KaK coaoxynaocrr,noaopo'roa na BceB03M02KHhIe yrJIhI, 0 ~ () ~ tt , aoxpyr- BceB03M02KHhIX ocea,3a,LIaHHhIX BCeMYl BeKTOpaMYl w e,LIYlHYl'IHOM crpepsr.
Coaoxynnocrr, acex TaKYlX BpaIIJ;eHYlM M02KHO onacar-s KaK map {w} pa,LIYlyca tt B TpexMepHoM eBKJIYl,LIOBOM npoc'rpaacrae. Ho na r-pannue aror-omapa Hy2KHO eIIJ;e OT02K,LIeCTBYlTb rrpOTYlBOrrOJI02KHhIe TO'IKYl, TaK KaK BpaIIJ;e
nae na yro.a tt aoxpyr aercropa w COBrra,LIaeT C BpaIIJ;eHYleM na yr-oa tt aoxpyraercropa -w (a ,LIpyrYlx COBrra,LIaIOIIJ;YlX BpaIIJ;eHYlM B HameM mape HeT).
3a~a~a. CYIIJ;eCTByIOT JIYl aBTOMOp¢Yl3MhI A: JR4 ---+ JR4 aJIre6phI KBa
TepHYlOHOB, T. e. npeoopaaoaaana (cKa2KeM, BeIIJ;eCTBeHHO-JIYlHeMHhIe), ,LIJIJI
KOTOphIX npa acex KBaTepHYlOHax x Yl y BhIrrOJIHJlIOTCJI COOTHOmeHYlJl
A(x + y) A(x) + A(y), A(xy) A(x)A(y) ?
3aMe~aHl'le. II:JIJI rrOJIJI KOMrrJIeKCHhIX 'IYlCM rpyrrrra aBTOMOP¢Yl3MOB
COCTOYlT Yl3 ,LIBYX ClJIeMeHTOB: A(z) z Yl A(z) z:KBaTepHYlOHHOe conpazceaae He J1BJIJleTCJI aBTOMOP¢Yl3MOM, TaK KaK
,LIJIJI JII06hIX KBaTepHYlOHOB z Yl W YlMeeT MeCTO JIerKO npoaepaexeoe T02K,LIeCTBO
ZW 'iiFz, a He z·'iiJ ('K6amep'liUo'li'lioe conpscncenue- «a'limUa6moMop(/ju3,M»).IIpYlMepOIvI aBTOMOp¢Yl3lVIa aJIre6phI KBaTepHYlOHOB J1BJIJleTCJI, O,LIHaKO,
Yl3MeHeHYle 3HaKOB d6YX Yl3 Tpex MHYlMhIX KOMrrOHeHT:
A(a+bi+cj+dk) a + bi - cj - dk.
3a~a~a. HaMTYl see aBTOMOp¢Yl3MhI aJIre6phI KBaTepHYlOHOB (OHYl no
X02KYl na ClTOT).
Hexo-ropsre IIpl'lMephI
Orrpezrezremre. Ilpeoopaaonaaxe (BeIIJ;eCTBeHHO rJIa,LIKoe) KOMrrJIeKCHOM
rrpOeKTYlBHOM rrJIOCKOCTYl na ce6J1 Ha3hIBaeTCJI nceedonooexmuewc»; eCJIYl
OHO nepeecmrr Ka2K,LIYIO KOMrrJIeKcHyIO rrpOeKTYlBHyIO npavyro B KOMrrJIeKC
HyIO rrpOeKTYlBHyIO npauyro.
3a~a~a. lI:oKa2KYlTe, 'ITO 6C,j/,'KOe nceedonpoexmuenoe npeotipasoeascue'KOMn,Jl,e'KC'liOU npOe'KmU6'1iOU nJWC'KOcmu na ceti« ,j/,6J1,,j/,emc,j/, Jl,u6o 'KOMnJl,e'KC'liblM npOe'KmU6'1ibIM npeotipcsoeawueu, Jl,u6o eeo npouseeoewuesc 'lia 'KOMnaexcnoe cotvpsocenue (51 He 3HaIO, BepHO JIYl ClTO 663 rrpe,LIrrOJI02KeHYlJl rJIa,LI
KOCTYl, HarrpYlMep - ,LIJIJI rOMeOMOp¢Yl3MOB).
20
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PaCCMOTpYlM TeTpaCl,LIp B eBKJIYl,LIOBOM TpeXMepHOM rrpOCTpaHCTBe. Ha
npaaneaas Yl3 ero IJ;eHTpa na BepIIlYlHhI Orrpe,LIMJlIOT 'IeThIpe TO'IKYl na Be
IIJ;eCTBeHHOM rrpOeKTYlBHOM rrJIOCKOCTYl JRp2. Tpyrma rrpOeKTYlBHhIX rrpeo6pa
30BaHYlM, COXpaHJlIOIl"\YlX ClTYl TO'IKYl, COBrra,LIaeT C rpynnoii A3 CYlMMeTpYlM
TeTpaCl,LIpa, COCTOJlIIJ;eM Yl3 24 opTorOHMbHhIX npeotipaaoaaaais eBKJIYl,LIOBa
rrpOCTpaHCTBa JR3,COXPaHJlIOIIJ;YlX TeTpaCl,LIp.
BJIo2KYlB JRp3 B KOMrrJIeKcHyIO rrpOeKTYlBHyIO rrJIOCKOCTb Cp2, MhI nony
'IYlM Yl TaM 4 TO'IKYl. L[To6hI «KoMrrJIeKCYl¢YlIJ;YlpOBaTb» rpyrrrry A3 , paccxeoTpYlM rpyrrrry acex COXpaHJlIOIl"\YlX CITy 'IeTBepKy rrCeB,LIOrrpOeKTYlBHhIX rrpe
06paClOBaHYlM KOMrrJIeKCHOM rrJIOCKOCTYl.
3a~a~a. lI:oKa3aTb, 'ITO TaK Orrpe,LIMeHHaJI «'1WMnJ/,e'KCu(/jU'Ka'/{u,J! zpynnb! A3 CUMMempuu mempooopo: ecmb zpynna B 3 scez 48 cUMMempuu O'Kmaodpa UJ/,U d6oucm6e'li'liozo O'Kma:Jdpy 'Ky6a 6 mpeXMep'liOM e6'KJ/,udo60M npocmpancmee. dTOT Ky6 M02KHO rrOJIy'IYlTb, ,LI06aBYlB K BepIIlYlHaM TeTpaCl,LIpa
C IJ;eHTpOM B HyJIe 'IeThIpe rrpOTYlBOrrOJI02KHhIX TO'IKYl.
3aMe~aHl'le.M02KHO Ha,LIeJlTbCJI, 'ITO 'K6amep'liUO'li'liOU eepcueil zpynnb!A 3 CUMMempuu mempaodp« (a paeno U 'KOMnJ/,e'KC'liOu eepcueit zpynnb! B 3
CUMMempuu 'Ky6a UJ/,U O'Kma:Jdpa) O'Ka;)ICemc,J! zpynna H 3 U3 120 cUMMempuuuxocaoopo: O,LIHaKO ClTO rrO,LITBep2K,LIaeTCJI TOJIbKO cTpaHHhIMYl ¢opMyJIaMYl
,LIJIJI 'IYlCM peoep: 3TYl 'IYlCJIa paaasr (pac. 10) 6 2 . 3 ,LIJIJI TeTpaCl,LIpa,
12 3·4 ,LIJIJI OKTaCl,LIpa (YlJIYl Ky6a) , 30 5·6,L1JIJ1 YlKOCaCl,LIpa (YlJIYl ,LIO,LIe
xaanpa).
IA3 1 = 24P=6
PMC. 10. Te rpaanp, OKTa3,l1p M MKOCa3,l1p, MX rpyIIIIhI cMMMeTpMM M 'fMCJIa pe
6ep P. CJIOBa TeTpa3,l1p, OKTa3,l1p, MKOCa3,l1p M ,l10,l1eKa3,l1p 03Ha'faIOT {('feThIpeX
rpaHHMK», {(BOCbMMrpaHHMK», {(,l1Ba,l1IIaTMrpaHHMK» M {(,l1BeHa,l1IIaTMrpaHHMK».
Bce ClTYl TpYl 'IYlCJIa YlMeIOT BYl,LI (n + l)(n + 2), r,LIe n 1, 2, 4pa3MepHOCTb (MHoroo6pa3YlJl BeIIJ;eCTBeHHhIX 'IYlCM, KOMrrJIeKCHhIX 'IYlCM Yl
KBaTepHYlOHOB, coo-rnerc'raeaao}.
rJIaBHaJI TPY,LIHOCTb YlCCJIe,LIOBaHYlJl KBaTepHYl3aIJ;YlYl COCTOYlT B OTCyT
CTBYlYl rOTOBOrO onpenezenas (pOJIb xor-opor-o npa KOMrrJIeKCYl¢YlKaIJ;YlYl Ylr
pan nepexon OT rrpOeKTYlBHhIX npeoopaaosaaais K rrCeB,LIOrrpOeKTYlBHhIM).
21
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M02KHO npennoaar-ar-s, 'ITO POJIb napsr TeTpa:'l,L\pOB, BrrYlCaHHbIX B Ky6
(Yl CBJl3aHHbIX C ,L\P06bIO IBsl/IAsl 48/24 2) ar-paer B KBaTepHYlOH
HOM CJIy'Iae narepxa Ky60B, BrrYlCaHHbIX B ,L\O,L\eKa:'l,L\p (peopasor Ka2K,L\OrO
Yl3 ClTYlX rrJlTYl Ky60B JlBJIJlIOTCJI HeKOTOpbIe Yl3 ,L\YlarOHaJIeM rrJlTYlyrOJIbHbIX
rpaHeM ,L\O,L\eKa:'l,L\pa, PYlC. 11). IIpYl CYlMMeTpYlJlX ,L\O,L\eKa:'l,L\pa 3TYl 5 Ky60B
rrepeCTaBJIJlIOTCJI, rrO,L\06HO TOMy, KaK ClTO rrpOYlCXO,L\YlT C napoii TeTpa:'l,L\pOB,
BrrYlCaHHbIX B Ky6, npa ,L\eMCTBYlYl r-pyrmsr 48 CYlMMeTpYlM Ky6a (coananaroIIJ;YlX C CYlMMeTpYlJlMYl ,L\BOMCTBeHHoro Ky6y orcraanpa, BepIIlYlHaMYl xo'ropor-oJlBJIJlIOTCJI IJ;eHTpbI rpaHeM xyoa}.
JJ
JJ
JJ
Ic--
Teopaa c-repeor-parpa-recxoii npoexnaa crpepsr na rrJIOCKOCTb npeapamaeTCJI npa nepexone OT KOMrrJIeKCHbIX 'IYlCM K KBaTepHYlOHaM (c HeYl36e2KHbIM
y'IeTOM ,L\BOMKYl Ponpar-eca) B 3aMe'IaTMbHYIO napaaeerpaaamoo xaarepHYlOHHOM rrpOeKTYlBHOM rrpJlMOM lHIp1 ~ 8
4KBaTepHYlOHal',<IYl, aHaJIOrYl'IHyIO
¢opMyJIaM «raar-eaca rrOJIOBYlHHOrO yrora»,
cos <p sirr <p2t
1 + t 2 't tg '£
2 '
rrapaMeTpYl3yIOIIJ;YlM OKPY2KHOCTb lRp1 Yl crpepy PYlIvIaHa CCP1 .
06 ClTOM KBaTepHYlOHHOM c-repeor-parpa-recxoii npoexmor Yl 06 ee npaJI02KeHYlJlX K YlCCJIe,L\OBaHYlIO CrrYlHOpHbIX ,L\BYJIYlCTHbIX HaKpbIBaIOIIJ;YlX Spin (4)
Yl Spirr(5) rpyrrrr 80(4) Yl 80(5) BpaIIJ;eHYlM crpep paauepaoc-reii 3 Yl 4 M02KHO
npo-recr-s B c'ramse: B. Jil. AP'liOJ/,bd, JIarpaH2KeB r-paccsraaaaa KBaTepHYlOH
HOrO rYlrrepCYlIvIIIJIeKTYl'IeCKOrO rrpOCTpaHCTBa (<f:>yHKIJ;. aHaJIYl3 Yl ero npaa.,T. 85, N~l (2001),74-77).
Yxaaaaasre ¢OpMyJIbI, KaK ClTO HYl CTpaHHO, so-rrepasrx, 6bIJIYl OTKpbITbI
npa pemeana 3HaMeHYlTOM 3a,L\a'IYl TeOpYlYl 'IYlCM: KaK HaMTYl ace «rrarpar-opOBbI TPOMKYl» IJ;MbIX 'IYlCM (X, Y, Z), JlBJIJlIOIIJ;YlXCJI ,L\JIYlHaMYl CTOpOH npa-MoyrOJIbHOrO TpeyrOJIbHYlKa, TaK 'ITO X 2 + y 2 Z2?
22
![Page 23: KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB CnHHOB V.I. _ Geometrija kompleksnyx chisel... · B. H. APHOl\bA vn~~~~ KOMnAEKCHblX '-IHCEA, KBATEPHHOHOB H CnHHOB H3j\ATEI\bCIllO MOCKOIICKOro](https://reader034.vdocuments.site/reader034/viewer/2022042318/5f071ec87e708231d41b67fb/html5/thumbnails/23.jpg)
IIpOCTeMIIHle IIpMMephI (32 + 42 52, 122 + 52 132) lVIHOrO MCIIOJIb30-
BaJIMCb B II:peBHeM ErMIITe, 'IT06hI CTpOMTb IIpJlMhIe yrJIhI, MCIIOJIb3YJl y3JIhI
na sepeaxe (HaIIpMMep, IIpM CTpOMTeJIbCTBe IIMpaMM,LI). Ho 06IIJ;aJI ¢OPMYJIa
6bIJIa aa ThICJI'IM JIeT ,LIO Ilarparopa oIIy6JIMKOBaHa, BMeCTe C TeopeMoM IIM
rparopa M C ,LIOKa3aTeJIbCTBaMM, na BaBMJIOHCKMX KJIMHOIIMCHhIX Ta6JIM'IKax
XaJI,LIeeB: KalK,LIaJI HecoKpaTMMaJI narparopoaa TpoMKa MMeeT BM,LI
r,LIe (u, v) - B3aMMHO npoc-rsre IJ;eJIhIe 'IMCJIa (pa3HOM 'IeTHOCTM, 'IT06hI
TpoMKa IIOJIy'IMJIaCb HeCOKpaTMMaJI).
C ,LIpyroM CTOpOHhI, Te LKe ¢OPMYJIhI MMeIOT TOIIOJIOrM'IeCKOe CO,LIepLKa
HMe, OIIMChIBaJI CTpyKTypy MHOLKeCTBa acex KOMIIJIeKCHhIX TO'IeK OKpyLKHO
CTM (T. e. KOMIIJIeKCHhIX peIIIeHMM ypaBHeHMJI OKpyLKHOCTM x 2 + y2 1 TaK Ha3hIBaeMOM pMMaHoBoM noaepxaocrx ,LIJIJI OKpyLKHOCTM). OHM LKe ,LIO
CTaBJIJlIOT, KaK MhI YBM,LIMM, TaKLKe yCJIOBMJI MHTerpMpyeMocTM B ClJIeMeH
TapHhIX ¢YHKIJ;MJlX TaK Ha3hIBaeMhIX «a6eJIeBhIX ,LIM¢¢epeHIJ;MaJIOB», BpO,LIe
~ dx (06IIJ;Me a6eJIeBhI MHTerpaJIhI - ClTO BCe MHTerpaJIhI BM,LIa
JR(x,y)dx B,LIOJIb KpMBOM H(x,y) 0, r,LIe H - lVIHOrO'IJIeH, R - pa
IJ;MOHaJIbHaJI ¢YHKIJ;MJI).
,1I)np<PepeHIJ;HaJIhHOe ypannerrae HhIOTOHa
II:M¢¢epeHIJ;MaJIbHOe upaenenue Hwomona
F(x),
OIIMChIBaIOIIJ;ee ,LIBMLKeHMe TO'IKM X e,LIMHM'IHOM MaCChI IIO IIPJlMOM IIO,LI ,LIeM
CTBMeM CMJIOBOrO IIOJIJI F, MMeeT nep6b!U unmezpaa ;:mepzuu
H(x,y) COIIst, r,LIe Hy22 + U(x);
y dx / dt - CKOpOCTb; U - IIOTeHIJ;MaJIbHaJI ClHeprMJI, OIIpe,LIeJIJleMaJI yCJIO-
BMeM F(x) -dU/dx.ECJIM F - lVIHOrO'IJIeH CTeIIeHM n, TO ypaaaenae 3aKOHa coxpaaenas
anepr-aa, H(x,y) E, OIIpe,LIeJIeH na «¢a30BOM IIJIOCKOCTM» C KOOp,LIMHaTaMM
X M Y aar-eopaa-recxyro KPMBYIO (rMIIepClJIJIMIITM'IecKYIO), 3aBMCJlIIJ;YIO OT «IIO
CTOJlHHOM ClHeprMM E», a BpeMJI t ,LIBMLKeHMJI B,LIOJIb Hee OIIpe,LIeJIJleTCJI a6e
JIeBhIM MHTerpaJIOM OT ,LIM¢¢epeHIJ;MaJIbHOM ¢OPMhI dt dx / y (IIOCKOJIbKy
Y dx/dt).
23
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COOTBeTCTBYIOII:\ee ClTOMY ,L\BYl2KeHYlIO ¢a30BOM TO'IKYl BeKTOpHOe none
na ¢a30BOM KpYlBOM H E eCTeCTBeHHO rrp0,L\OJI2KaeTCJI na BCIO KOMrrJIeKC
HyIO pYlIvIaHOBy rrOBepXHOCTb (TaK, 'ITO dt 1 na sex'ropax rroJIJI). Y,L\YlBYl
TMbHbIM ¢aKTOM OKa3bIBaeTCJI TO, 'ITO ClTO BeKTOpHOe none onacsraaer ,L\BYl
2KeHYle no PYlMaHOBOM rrOBepXHOCTYl 3aIIOJIHJlIOII:\eM ee «HeC2KYlMaeMOM 2KYl,L\KO
CTYl» (YlMeeT «,L\YlBepreHIJ;YlIO HOJIb», 'ITO B ,L\pyrYlX TepMYlHax oanaxae'r «aaMKHyTOCTb» ¢OPMbI dt, T. e. TO, 'ITO OHa OKa3bIBaeTCJI JIOKaJIbHO rrOJIHbIM
,L\Yl¢¢epeHIJ;YlaJIOM) .
3a~a~a. IIYCTb rrOTeHIJ;YlaJIbHaJI ClHeprYlJl U - lVIHOrO'IJIeH 'IeTBepToM
creneaa C ,L\BYMJI MYlHYlMYMaMYl (<,L\ByMJI rrOTeHIJ;YlaJIbHbIMYl J1MaMYl»). Pac
CMOTpYlM nepnomrsecrcae ,L\BYl2KeHYlJl B TOM Yl B ,L\pyroM J1Me, C O,L\YlHaKOBbIMYl
3Ha'ieHYlJlMYl rrOCTOJlHHOM rrOJIHOM anepr-aa, E. Crrpanraaaerca, nepnon KO
Toporo Yl3 HYlX 60JIbIIle: ,L\BYl2KeHYlJl B 60JIee rJIy60KoM J1Me YlJIYl B MeHee rJIy
60KOM?
OTBeT. ::::ha aanaxa - TOrrOJIOrYl'IeCKaJI. 06a nepnona O,L\YlHaKOBbI, no
CKOJIbKY pYlMaHOBa noaepxaocri, ¢a30BOM KpYlBOM H E - TOp, nepaonn
'IeCKYle ,L\BYl2KeHYlJl - ,L\Ba ero MepYl,L\YlaHa, a rrOTOKYl HeC2KYlMaeMOM 2KYl,L\KOCTYl
na Tope -repea JII06bIe ,L\Ba MepYl,L\YlaHa O,L\YlHaKOBbI.
3aMe~aHHe. IIHTerpaJIbI ,L\Yl¢¢epeHIJ;YlaJIbHOM ¢OPMbI dt dx/y no
BceB03M02KHbIM 3aMKHYTbIM rryTJlM na Tope, BbIXO,L\J1II:\YlM Yl3 O,L\HOM Yl TOM
2Ke TO'IKYl, 06Pa3YIOT «peIIleTKY»: aoeneay rpyrrrry Z,Wl + Z,W2 T, r,L\e Wl
Yl W2 - mrrer-pansr B,L\OJIb napaanena Yl B,L\OJIb MepYl,L\YlaHa Topa. 3Ha'ieHYle tmrrer-paxa no BbIXO,L\J1II:\YlM Yl3 BbI6paHHOM TO'IKYl He3aMKHYTbIM rryTJlM J1BJIJI
eTCJI MHOr03Ha'iHOM ¢YHKIJ;YleM OT KOHe'IHOM TO'IKYl Ha Tope, rrpYl'IeM Bce
3Ha'ieHYlJl ClTOM MHOr03Ha'iHOM ¢YHKIJ;YlYl B TO'IKe rrOJIY'IaIOTCJI Yl3 O,L\HOrO Yl3
HYlX rrpYl6aBJIeHYleM acex 'IYlCM Yl3 pememca f.TaKYlM 06pa30M, casra nama TOpYl'IeCKaJI pYlMaHOBa noaepxaoc-rs npen
CTaBJIJleT C060M (C TO'IHOCTbIO ,L\O KOMrrJIeKCHOrO ,L\Yl¢¢eOMOp¢Yl3Ma) rpax
Top-rrpOCTpaHCTBO C/f C/(Z,Wl + Z,W2), r,L\e KOMrrJIeKCHOe 'IYlCJIO ,\ ~~
He BeII:\eCTBeHHO.
BbI6YlpaJI pa3JIYl'IHbIe MHOrO'IJIeHbI 'IeTBepTOM crenena, M02KHO nony
'IYlTb, 'ITO ace HeBeII:\eCTBeHHbIe 3Ha'ieHYlJl '\2 Yl ace TOpYl'IeCKYle PYlMaHOBbI
rrOBepXHOCTYl ,L\OCTaBJIJlIOTCJI ClTOM KOHCTpyKIJ;YleM ('ITO ,L\OKa3aTb Y2Ke He TaK
JIerKO).
Bce crpepaxecxae pYlMaHOBbI rrOBepXHOCTYl KOMrrJIeKCHO ,L\Yl¢¢eOMOp¢HbI
CTaH,L\apTHOM ccpepe PYlMaHa 8 2 Cpl.BBYl,L\y CTOJIb 60JIbIIlOM Ba2KHOCTYl ClTYlX BorrpOCOB, 51 CKa2Ky 0 HYlX He
CKOJIbKO CJIOB.
24
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OT TeOpeMhI IIl'Ieparopa K pl'IMaHOBhIM IIOBepXHOCTHM
PaCCMOTpMM OKpyLKHOCTb X 2 + y2 1 M 6Y,LIeM CHa'Ia.JIa MCKaTb na HeM
TO'IKM C paIJ;MOHa.JIbHbIMM KOOp,LIMHaTaMM (x,y). O,LIHa TaKaH TO'IKa M3
BeCTHa: (x 1, yO). IIpOBe,LIeM -repea :'lTy TO'IKy rrpmeyto C TaHreHCOM
yrora HaKJIOHa t K OCM OX (T.e. C ypaaaeaaev y t(x -1), pMC. 12).
y
PMC. 12. Ilocr-poeaae pal\MOHaJIbHOM IIapaMeTpM3al\MM OKpy:lKHOCTM TaHreH
COM t IIOJIOBMHHOro yraa,
O,LIHa TO'IKa nepecexenas :'lTOM rrpJlMOM C OKpyLKHOCTbIO HaM yace M3
BeCTHa. IIO:'lTOMy BTOPOM xopeas x rrOJIY'IaIOIIJ;erOCJI npa ¢MKCMpOBaHHOM tKBa,LIpaTHOro ypaBHeHMJI ,LIJIJI KOOp,LIMHaTbI x TO'IKM rrepeCe'IeHMJI rrpJlMOM
C OKpyLKHOCTbIO BbIpaLKaeTCJI -repea 3Ha'IeHMe napasee'rpa rrpJlMOM t paIJ;MO
Ha.JIbHO. CJIe,LIOBaTeJIbHO, OKpyLKHOCTb - «PaIJ;MOHa.JIbHaH KpMBaH», ,LIOrrYCKa
IOIIJ;aH rrapaMeTpM3aIJ;MIO
x P(t), Y Q(t),
r,LIe P M Q - paIJ;MOHa.JIbHbIe ¢YHKIJ;MM.
IIpOBO,LIJl Te LKe BbI'IMCJIeHMJI JlBHO, MbI JIerKO HaxO,LIMM, 'ITO
(1)
P Q2t
-1 + t 2 '
IIPM paIJ;MOHa.JIbHbIX 3Ha'IeHMJlX x My 'IMCJIO t ~ paIJ;MOHa.JIbHO, a npa pa
IJ;MOHa.JIbHOM 3Ha'IeHMM t MbI HaxO,LIMM PaIJ;MOHa.JIbHbIX x M y no ¢opMyJIaM (1).IIPM -t u/v C IJ;eJIbIMM U M V MbI HaXO,LIMM YKa3aHHbIe BbIII1e ¢OPMYJIbI acex
«IIM¢arOpOBbIX» TpoeK (r,LIe x X / Z, y Y / Z). lIcCJIe,LIoBaHMe necoxpa
TMMOCTM (B3aMlVIHoM npocro-rsr) HeCJIOLKHO (He Ha,LIO, 'ITo6bI U M v o6a 6bIJIM
He'IeTHbIMM!) .
4-KOMII.'f. 25
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C ,LIpyroM CTOpOHhI, IIOBepXHOCTb, 06pa30BaHHaH '/W.M,nJl,e'KC1ibl.M,U peme
HYlJlMYl ypaaaenas X 2 + y2 1, BKJIIO'IaH «6ecKOHe'IHo-y,LIaJIeHHhIe», OKa3hIBa
eTCJI, KaK ClTO IIOKa3hIBaeT IIapaMeTpYl3aLJ;YlJl KOMIIJIeKCHhIM IIapaMeTpOM t,c¢epOM PYlMaHa, 8 2 CCP1.
PMC. 13. 3JIJIMIITM'IeCKaJI KpMBaJI CTeIIeHM 3: ee BeII\eCTBeHHhIe TO'IKM M ee
pMMaHOBa IIOBepXHOCTb.
II:JIJI ,LIpyrYlx lVIHOrO'IJIeHOB H (X, y) IIOJIY'IYlJIYlCb 6hI ,LIpyrYle IIOBepXHOCTYl
H 0, KOTOphIe MoryT Yl He 6hITb crpepaxor. Hanpmrep, YPaBHeHYle «ClJIJIYl
IITYl'IeCKOM KPYlBOM»
y2 x3 - X + E
3a,LIaeT IIpYl IIO'ITYl acex E IIOBepXHOCTb Topa 8 1 x 8 1 , Ha3hIBaeMOrO TaK2Ke
«c¢epoM C O,LIHOM py'IKOM» (pac. 13).
PMC. 14. PMMaHoBa IIOBepXHOCTb poaa g 3: crpepa C TpeMJI py'IKaMM.
B 06IIJ;eM CJIy'Iae (aanpaxcep, IIpYl aaxreae IIOKa3aTeJIJI 3 B x3 60JIee BhI
COKYlM IIOKa3aTeJIeM 29 +1) IIOJIy'IaeTCJI B Ka'IeCTBe pYlMaHOBOM IIOBepXHOCTYl
crpepa C 9 py-ncasor (pac. 14, 'IYlCJIO 9 Ha3hIBaeTCJI pOaO.M, IIoBepXHOCTYl).
Beci, Ha60p CBJl3HhIX 3aMKHYThIX rJIa,LIKYlX opaem-apyeszsrx IIOBepxHO
CTeM (663 xpaea YlJIYl oco6eHHocTeM) YlC'IepIIhIBaeTCJI C¢epaMYl C 9 py-ncasor
(9°,1,2, ... ).
26
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OCHOBHaH TeOpeMa TeOpMM MHTerpMpOBaHMJI a6MeBhIX ,LIM¢¢epeHIJ;Ma
JIOB B,LIOJIb aJIre6paM'IeCKMX KpMBhIX H(x,y) 0 COCTOMT B TOM, 'ITO 6Cema'/\:ue unsneepaa« (om npOU360Jl,b'lib1X pa'U,UO'liaJl,b'libIX ¢OPM R(x, y) dx) 6epymcJt 6 cJJl,eMe'limap'libIX ¢Y'li'/\:'U,UJtX ecsni U mOJl,b'/\:O ecsni pod pUMa'li060Unoeepimoctmi ,/\:PU60U H 0 paeew 'liyJl,'IO (m. e. ecsni ona du¢¢eoMop¢'liac¢epe).
HaIIpMMep, ,LIJIJI H x2 + y2 -1 ace TaKMe MHTerpaJIbI 6epyTcJI, TaK KaK
OHM CBe,LIeHbI HaMM BbIIIle K MHTerpaJIaM OT paIJ;MOHaJIbHbIX ¢YHKIJ;MM OT IIe
peMeHHOM t
JR(x,y)dx JR(P(t),Q(t))P'(t)dt
(CITO - «TeOpMJI IIO,LICTaHOBOK dMJIepa», TOIIOJIOrM'IeCKMM CMbICJI KOTOPOM
OT cTy,LIeHTOB BCer,LIa cKpbIBaIOT).
KaLK,LIaH KpMBaH pona 0 paIJ;MOHaJIbHa, T. e. ,LIOIIycKaeT paIJ;MOHaJIbHyIO
IIapaMeTpM3aIJ;MIO, aHaJIOrM'IHyIO HaM,LIeHHOM HaMM ,LIJIJI OKpyLKHOCTM JlBHO
(,LIOKa3aTb CITy napasrerpaayesroc-rt, He O'IeHb Tpy,LIHO, HO ace LKe TyT Tpe
6yeTcJI nexoropoe BJIa,LIeHMe KOMIIJIeKCHOM reOMeTpMeM).
ECJIM LKe P0,LI pMIvIaHOBOM IIOBepxHOCTM KpMBOM H 0 60JIbIIle HyJIJI,
TO MHTerpaJI OT IIO,LIXO,LIJlIIJ;eM paIJ;MOHaJIbHOM ,LIM¢¢epeHIJ;MaJIbHOM ¢OPMbI
B,LIOJIb ClTOM KpMBOM OKa3bIBaeTCJI MHOr03Ha'iHOM ¢YHKIJ;MeM B KOMIIJIeKCHOM
06JIaCTM CO CTOJIb CJIOLKHbIM BeTBJIeHMeM, KaKOrO He MOLKeT MMeTb HMKaKaH
ClJIeMeHTapHaH ¢YHKIJ;MJI. IIoClTOMY MHTerpaJIbI B ClTOM CJIy'Iae He BCer,LIa CBO
,LIJlTCJI K ClJIeMeHTapHbIM ¢YHKIJ;MJlM.
IIpOCTeMIIleM 3a,LIa'IeM TaKOrO pona JlBJIJleTCJI 3a,LIa'Ia 0 BbI'IMCJIeHMM ,LIJIM
HbI ,LIyrM ClJIJIMIICa, CBO,LIJlIIJ;aHCJI KaK paa K MHTerpMpOBaHMIO B,LIOJIb KpMBOM
pona 9 1 (KOTOPaH IIOClTOMy-TO M Ha3bIBaeTCJI cJJl,JI,unmu"iec,/\:ou '/\:PU60U).TaKMM 06pa30M, HaIIIa TeOpMJI CBJl3bIBaeT KOMIIJIeKCHbIM aHaJIM3 M C TO
IIOJIOrMeM, M C TeopMeM 'IMCM, M C TeopMeM aJIrOpMTMM'IeCKOM pa3peIIlMMOcTM
3a,LIa'IM MHTerpMpOBaHMJI.
3aMe~aHl'le. Taxoe LKe TOIIOJIOrM'IeCKOe ,LIOKa3aTMbCTBO MMeeT M Te
opexea A6MJI 0 HeB03MOLKHOCTM peIIleHMJI B pa,LIMKaJIax otinrer-o ypaBHeHMJI
IIJlTOM (MJIM 60JIee BbICOKOM) cTeIIeHM, HaIIpMMep, yace ypaBHeHMJI
x5 + ax + 1 0
(K KOTOpOMy, BIIpO'IeM, CBO,LIJlTCJI ace ypaBHeHMJI IIJlTOM cTeIIeHM). 3,L1eCb
HyLKHO MCCJIe,LIOBaTb BeTBJIeHMe IIJlTM3Ha'iHOM KOMIIJIeKCHOM ¢YHKIJ;MM x ap
ryMeHTa a, 3a,LIaHHOM BbIIIMCaHHbIM ypaaaenaexe, KOr,LIa KOMIIJIeKCHaH nepe
MeHHaH a 06XO,LIMT aoxpyr TO'IeK BeTBJIeHMJI ClTOM IIJlTM3Ha'iHOM ¢YHKIJ;MM.
4* 27
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BeTBAeHMe ¢YHKQMM OrrMCMBaeTCH rrepeCTaHOBKaMM rrHTM AOKaAbHMX
AMCTOB X J (a), M rrepeCTaHOBKM, COOTBeTcTByIOIIJ;Me BceB03M02KHMM rrYTHM,
He rrpOXO,LIHIIJ;MM -repea TO'IKy BeTBAeHMH, 06pa3yIOT rpyrrrry Ha3MBaeMyIO
«r-pynnoii MOHO,LIPOMMM» MHOr03Ha'IHOM ¢YHKQMM x OT apr-ysrem-a a, pMC. 15.
o
PMC. 15. Tpynrra MOHO,l1pOMMM cPyHKrrMM «KBa,l1paTHhIM xopem,» COCTOMT M3
,l1BYX rrepeCTaHOBOK 060MX xopaea.
Tpyrma MOHO,LIpOMMM AI060M KOM6MHaQMM pa,LIMKaAOB paapemmra (CBO
,LIMTCH K KOCOMy rrpOM3Be,LIeHMIO KOMMyTaTMBHMx rpyrrrr). A rpyrrrra MOHO
,LIpOMMM YKa3aHHoM BMIIIe rrHTM3Ha'IHOM ¢YHKQMM CO,LIep2KMT ace nepecra
HOBKM 5 ClAeMeHTOB M no-touy Hepa3peIIIMlVIa. 3Ha'IMT Hepa3peIIIMMO B pa
,LIMKaAaX M Harne ypaaaemre (,LIa2Ke eCAM ,LI06aBMTb K pa,LIMKaAaM ace O,LIHO
3Ha'IHMe ¢YHKQMM). 06 ClTOM TeopeMe A6MH M02KHO npo-orram, rrO,LIp06Hee
B KHMre B. B. Aaexceena «Teopeua A6MH B aanaxax M peIIIeHMHX»(M.: Ha
yxa, 1976), BocrrpOM3BO,LIHIIJ;eM MOM AeKQMM IIIKOAbHMKaM 1964 rO,LIa.
EMM ypaaaeaae KpMBOM H(x,y) 0 3a,LIaHO, TO HaMTM ee pO,LI 9 He TaK
Y2K AerKO. 3,L1eCb nosror-aer C006pa2KeHMe «MTaAbHHCKOM aAre6paM'IeCKOM
reOMeTpMM»: eCAM crenem, n MHOrO'IAeHa H ¢MKCMpOBaHa, TO pod 6ydemodwusc U me.M 'JICe dJ1,J! nownni scez 3'1ia"ie'liUU '1W3¢¢uv,ue'lim06 unozowaewa.11:60 Te cnenaansmae MHOrO'IAeHM, ,LIAR KOTOpMX pO,LI MHOM, pe,LIKM. IIo,LI06HO
TOMy, KaK Te 3Ha'IeHMH napaxeerpa C, ,LIAR KOTOpMX ypaanemre xn C MMeeT
HeTMrrM'IHOe, TO eCTb OTAM'IHOe OT n, 'IMMO xopnea, cnenaansmae MHOrO
'IAeHM 06pa3yIOT MH02KeCTBO, 3a,LIaBaeMOe KOMrrAeKCHMM ypaaaexaev (<,LIMC
KpMMMHaHT paaea HyAIO» ), T. e. ,LIByMH BeIIJ;eCTBeHHMMM ypaBHeHMHMM (B yxa
3aHHOM BMIIIe npmeepe - C a + bi 0 03Ha'IaeT cMcTeMy (a 0, b 0)).IIwToMy 'liecnev,uaJl,b'lible .M'liOZO"iJl,e'libl H otipasinom 6 npocmponcmee
scez .M'liOZO"iJl,e'li06 da'li'liou cmenenu n C6J!3'1iOe .M'liO'JICeCm60, npa ,LIBM2KeHMM
B,LIOAb xo'roporo TOrrOAOrM'IeCKMe MHBapMaHTM (aarrpauep, PO,LI) COOTBeT
cTByIOIIJ;eM ,LIBM2KYIIJ;eMCH TO'IKe pMMaHOBOM rrOBepxHOCTM OCTaIOTCH HeM3MeH
HMMM.
OCTaeTCH HaMTM pO,LI 9 ,LIAH od'liozo HeCrreQMaAbHOrO npmrepa. dTO
He TaK Y2K Tpy,LIHO C,LIMaTb: rO,LIMTCH, nanpmeep, rrO'ITM BMp02K,LIeHHaH KpM-
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BaH H 0, r,[le H L 1 · .... L n + e, M r,[le L J - JIMHeMHhIe HeO,[lHOp0,[l
HhIe ¢YHKIJ;MM oomero rrOJIO.lKeHMJI. ::::ha pMlvIaHOBa rrOBepXHOCTb rrOJIy'IaeTCJI
M3 n crpep ,[I06aBJIeHMeM N Tpy6o'IeK, r,[le N - 'IMCJIO TO'IeK nepece-renas
n npnxnax (pMC. 16).
PMC. 16. TOIIOJIOrMJI BeII\eCTBeHHoM anreopaa-recxoa KpMBOM M pMMaHoBoM IIO
BepXHOCTM IIO'fTM BhIpO:lK,l1eHHOM KpMBOM CTeIIeHM 3 (,l10 B03MYII\eHMJI 'fMCJIO
crpep n 3, 'fMCJIO COe,l1MHJIIOII\MX -rpyoo-rex N 3, PO,l1 g 1).
9(n - l)(n - 2)
2
L[To6hI ee 3arrOMHMTb, 3aMeTMM, 'ITO P0,[lhI KpMBhIX creneaa n 1 (npaxnax)
M c'reneaa n 2 (3JIJIMrrCOB, r-anepfio,a, naparion] paaasr HyJIIO, T. e. 3TM KpM
BhIe ,[IOrrycKaIOT pauaoaam.asre napasserpaaauaa, a MX pMMaHOBhI nosepx
HOCTM ,[IM¢¢eOMop¢HhI crpepai«.
Pau;MOHaJIbHhIe KpMBhIe Ha3hIBaIOTCJI eme Y'liu",;ypCaJWlibl.M,U, T. e. O,[lHO
npofier-aexrsnra, TaK KaK Ka.lK,[IyIO M3 HMX MO.lKHO Ha'IepTMTb (B BeII:\eCTBeH
HOM CJIy'Iae) O,[lHMM pOC'IepKOM nepa, He OTphIBaH ero OT 6ylVIarM (3,[1eCb
Hy.lKHO MJIM MMeTb B BM,[Iy KPMBYIO na npOe",;mU6'1iOU rrJIOCKOCTM, MJIM .lKe
npenrrcaar-arr, BeII:\eCTBeHHYIO 'IaCTb KpMBOM B a¢¢MHHOM rrJIOCKOCTM "';O.M,
na",;m'liou: 3JIJIMrrC yHMKypCaJIeH, a a¢¢MHHaH 'IaCTb r-anepfionsr - HeT).
lI:oKa3aTeJIbCTBO yHMKypCaJIbHOCTM paIJ;MOHaJIbHOM KpMBOM rrOJIy'IaeTCJI
rrpOCTO npa paCCMOTpeHMM BeII:\eCTBeHHhIX 3Ha'IeHMM paIJ;MOHaJIbHO napaveTpM3yIOII:\eM KPMBYIO nepeueaaoii t.
ECJIM MCXO,[lHaH KpMBaH H 0 He rvramcaa, TO Ka.lK,[IaH ee oco6eHHOCTb
YMeHbIIlaeT PO,[l COOTBeTCTBYIOII:\eM pMMaHOBOM rrOBepXHOCTM (aa KOTOPOM,
HarrpMMep, TO'IKe rrpOCToro caMorrepeCe'IeHMJI MCXO,[lHOM KpMBOM COOTBeT
CTByIOT ,[IBe TO'IKM «aa pa3HhIX JIMCTax»). OKa3hIBaeTCJI, KpMBaH c'reneaa n
29
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C (n-l )2( n-2) OC06hIMYl TO'IKaMYl (a 60JIbIIle YlX Yl He 6hIBaeT) YlMeeT Y2Ke P0,L\ 0,T. e. OHa paaraoaaasaa Yl ynaxypcam.na, a mrrer-pama OT paIJ;YlOHaJIbHhIX
¢OPM B,L\OJIb Hee BhI'IYlCJIYlMhI B ClJIeMeHTapHhIX ¢YHKIJ;YlJlX. TaKOBa, K npa
Mepy, «BhIp02K,L\eHHaJI ClJIJIYlrrTYl'IeCKaJI KpYlBaJI» (pac. 17) y2 x3 - 3x + 2(c OC060M TO'IKOM npocror-o caaeonepece-reaas xl,yO).
P0,L\: 9 = 0
c'renem.: n = 3
cO - @)R: ()Z - 0-\I ,
9=0 9=1PMC. 17. YHMKypCaJIbHaJI ocooaa KpMBaJI, B,l10JIb KOTOPOM 6epyTCJI see a6eJIeBbI MHTerpaJIbI, M ee MaJIaJI rrepec-rpoinca B neocooyro 3JIJIMIITM'fecKYIO KpM
ByIO (B BeII\eCTBeHHoM M B KOMIIJIeKCHOM 06JIaCTM).
3aMe~aHl'le. B Ka'IeCTBe napasee'rpa t, -repea KOTOphIM KOOp,L\YlHaThI
x Yl y TO'IeK HaIIIeM YHYlKYPCaJIbHOM KpYlBOM BhIpa2KaIDTCJI pau;YlOHaJIbHO,
M02KHO B3J1Tb TaHreHC yrora HaKJIOHa K OCYl Ox rrpJlMOM, COe,L\YlHJlIDII:\eM nay
'IaeMyID TO'IKy KpYlBOM C TO'IKOM casronepece-reaaa (npoaena-re BhI'IYlCJIeHYlJl
x(t) Yl y(t) - OHYl rr03BOJIJlT BhI'IYlCJIJlTb BCe a66JIeBhI mrrer-pansr B,L\OJIb na
IIleM KPYlBOM).
Me2K,L\y npo-noa, ara KOHCTpyKIJ;YlJl rrOJlCHJleT, KaK na pYlMaHOBOM no
BepXHOCTYl KpYlBOM BMeCTO TO'IKYl caMorrepeCe'IeHYlJl YlCXO,L\HOM KpYlBOM rrO
JlBJIJlIDTCJI ,L\Be 06hIKHOBeHHhIe TO'IKYl. II:Ba 3Ha'IeHYlJl h Yl t2 napaxeerpa t,COOTBeTCTBYIDII:\Yle TO'IKe casronepece-reaas YlCXO,L\HOM KpYlBOM, - ClTO TaH
reHChI HaKJIOHa yrJIOB HaKJIOHa K OCYl x o6eYlx rJIa,L\KYlX BeTBeM YlCXO,L\HOM
KpYlBOM, npoxonsmax -repea ee TO'IKy caaeorrepecexeaas.ITepexo,L\ OT KpYlBOM C caMorrepeCe'IeHYlJlMYl K ee rJIa,L\KOM PYlMaHOBOM
rroBepXHOCTYl, na KOTOPOM TO'IKa caaeonepece-reaas npencraaneaa HeCKOJIbKO
paa, Ha3hIBaeTCJI 'li0p.MaJl,U3av,ueu YlCXO,L\HOM KpYlBOM.
OKa3hIBaeTCJI, BCJlKaJI aJIre6paYl'IeCKaJI KPYlBaJI M02KeT 6hITb anreopaa
'IeCKYl HOpMaJIYl30BaHa, T. e. M02KeT 6hITb nosy-rena Yl3 nexo-ropoii rJIa,L\KOM
30
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am-eopaa-recxoii Kpl'lBOM (M3 CBOeM pMMaHOBOM rrOBepXHOCTM) aareopaa-re
CKMM oTo6pa2KeHMeM (oTo6pa2KeHMeM, KOTOpOe, npaana, M02KeT rrOChLJIaTb
B HeKOTOpbIe - OC06bIe - TO'IKM KpMBOM-06pa3a HeCKOJIbKO pa3HbIX TO'IeK
pMMaHOBOM rrOBepXHOCTM).
3a~a~a. HOPMa.rrM30BaTb JIeMHMcKaTy y2 X 2 - X 4. YHMKypCa.rrbHa JIM
ClTa KpMBaH?
MaTeMaTI'I~eCKl'leTP0I'IIJ;hI
TPM BapMaHTa - BeII:\eCTBeHHbIM, KOMrrJIeKCHbIM M KBaTepHMOHHbIM
MMeIOT MHOrMe MaTelVIaTM'IeCKMe TeOpMM, rrpM'IeM e,LIMHCTBO KaK COOTBeT
CTBYIOII:\MX TeOpeM, TaK M MX rrpMJI02KeHMM (TO B TOrrOJIOrMM, TO B ¢M3MKe,
TO B TeOpMM 'IMCM MJIM B a.rrre6pe) pacnoaaar-s He BCer,LIa JIerKO.
IIpMBe,LIy JIMIIlb HeCKOJIbKO npmeepoa".Hpaxrep 1. COBrra,LIeHMe rrpoeKTMBHOM rrpJlMOM C OKPY2KHOCTbIO:
KOlVIIIJIeKCM¢MKaLJ;MeM ClTOrO ¢aKTa OKa3bIBaeTCJI y,LIMBMTMbHaH Teope
Ma IIoHTpJlrMHa (OTKPbITaH MM B TpM,LIU;aTbIe rO,LIbI, HO He orry6JIMKOBaHHaH,
M norowy M3BeCTHaH na aanane no.n MMeHaMM Tex MaTeMaTMKOB, xo'ropsre
B IIleCTM,LIeCJlTbIe rO,LIbI orry6JIMKOBa.rrM CBOM ,LIOKa3aTMbCTBa B OTBeT na MOM
aorrpoc, M3BeCTHO JIM MM ,LIOKa3aTMbCTBO ClTOM TeopeMbI IIoHTpJlrMHa).
Teopexra. etJa",;mopnpocmpancm60 "';OMnJl,e"';Cnou npoe",;mU6nou nnoc
",;ocmu no ee 6ew,ecm6ennoMy du¢¢eOMop¢U3MY «"';OMnJl,e"';Cnoe conpJI/JlCe
nue: du¢¢eOMop¢en "iemb!peXMep'/iOU cfiepe:
TaKMM 06pa30M, npa KOMrrJIeKCM¢MKaLJ;MM pa3MepHOCTb e,LIMHMU;a npoex
TMBHOrO rrpOCTpaHCTBa 3aMeHMJIaCb ,LIBOMKOM, M B,LIo6aBOK rrpMIIlJIOCb ¢aKTO
pM30BaTb no rpyrrrre aBTOMOP¢M3MOB rrOJIJI KOMrrJIeKCHbIX 'IMCM ('ITO M02KHO
6hLJIO 6bI ,LIMaTb M B BeII:\eCTBeHHOM CJIy'Iae, r,LIe, O,LIHaKO, e,LIMHCTBeHHbIM as
TOMOP¢M3M - T02K,LIeCTBeHHoe npeotipaaoaanae}.
* BOJIee rr0,l1p06HOe oocyxcneaae 60JIbIIIerO aaoopa c!>aKTOB MO:lKHO HaMTM B cra-rr,e:B. N. ApnO.f1,bO, «IIoJIMMaTeMaTMKa: JIBJIJIeTCJI JIM MaTeMaTMKa e,l1MHOM HayKoM MJIMHa60pOM peMeCeJI» B C60pHMKe «Mathematics: Frontier and Perspectives», M3,l1aHHOMMe:lK,l1yHapO,l1HbIM MaTeMaTM'feCKMM COI030M M AMepMKaHcKMM MaTeMaTM'feCKMM 06II\eCTBOM B 2000 r-ozty B 03HaMeHOBaHMe KOHIIa TbICJI'feJIeTMJI (non pen. B. H. ApHOJIb,l1a, M. ATbJI, II. Jlaxca M B. Maaypa). IIJIaHMpyeMbIM pyCCKMM nepeaozt 06'hJIBJIeH M3,l1aTeJIbCTBOM «Cl>a3MC», J\10cKBa.
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Yra,LIaTb KBaTepHYlOHHbIM aHMOr rrpe,LIbI,LIYII:\eM TeopeMbI H6JIerKO,
HO aHMYl3 paayxoror-o ee ,LIOKa3aT6JIbCTBa noxaasraae'r CJIe,LIyIOII:\ee*:
lHIp4/ Aut / Conj ~ 8 13
.
3,L1eCb Ha'IYlHaTb rrpYlXO,LIYlTCJI C KBaTepHYlOHHO 'IeTbIpeXMepHoro (T. e. Be
II:\eCTBeHHO IIleCTHa,LIu;aTYlMepHoro) rrpOeKTYlBHOrO rrpOCTpaHCTBa, a ¢aKTO
pYl30BaTb - no TpexMepHOM (Yl30MOP¢HOM 80(3)) rpyrrrre aBTOMOP¢Yl3MOB
Yl eII:\e no aHTYlaBTOMOP¢Yl3MY KBaTepHYlOHHOrO COrrpJl2KeHYlJl.
IIoy'IYlT6JIbHO, O,LIHaKO, 'ITO ,LIOKa3aT6JIbCTBO acex Tpex rrepe'IYlCJIeHHbIX
BbIII1e ¢aKTOB COBepIIleHHO rrapMJI6JIbHbI - Hy2KHO TOJIbKO 3aMeHJlTb Be
II:\eCTBeHHbIe 'IYlCJIa na KOMrrJIeKCHbIe (Yl KBaTepHYlOHbI), a KBa,LIpaTYl'IHbIe
¢OPMbI (xo-ropsae B BeII:\eCTBeHHOM CJIY'Iae 3aIIYlCbIBaIOTCJI B rrO,LIXO,LIJlII:\eM
CYlCTeMe KOOp,LIYlHaT B BYl,LIe 2:;;'=1 amx;,,) - na BeII:\eCTBeHHbIe ClpMYlTOBbI
(Yl r-anepapxorroasr] ¢OPMbI, xo'ropsre 3aIIYlCbIBaIOTCJI B TaKOM 2Ke BYl,LIe C aa
MeHOM TOJIbKO KBa,LIpaTOB x;" na KBa,LIpaTbI MO,LIyJIeM Ix;"l.Orrpe,LI6JIeHYle ClPMYlTOBbIX Yl rYlrrepClpMYlTOBbIX ¢OPM (B KOMrrJIeKCHOM
BeKTopHOM rrpOCTpaHCTBe en Yl KBaTepHYlOHHOM lHIn ) - ClTO 06bI'IHbIe Be
II:\eCTBeHHbIe KBa,LIpaTYl'IHbIe ¢OPMbI (B COOTBeTCTBYIOII:\eM JR2n YlJIYl JR4n) ,YlHBapYlaHTHbIe OTHOCYlT6JIbHO YMH02KeHYlJl BeKTopoB-apryMeHToB na KOM
rrJIeKCHbIe 'IYlCJIa (KBaTepHYlOHbI) HOpMbI e,LIYlHYlU;a.
reOIvleTpYl'IeCKYlM 06'heKT, COOTBeTCTBYIOII:\YlM rrOJI02KYlT6JIbHO-Orrpe,LIe
JIeHHOM KBa,LIpaTYl'IHOM ¢opMe I, - ClTO ClJIJIYlrrCOYl,LI f 1. TaK 'ITO Clp
MYlTOBbIM (a B KBaTepHYlOHHOM CJIY'Iae - rYlrrepClpMYlTOBbIM) ¢OPlVIaM CO
OTBeTcTByIOT ClJIJIYlrrCOYl,LIbI BpaII:\eHYlJl CO CrreU;YlMbHbIMYl CYlMMeTpYlJlMYl: OHYl
nepexonar B ce6J1 npa YMH02KeHYlYl acex nercr-opoa rrpOCTpaHCTBa ClJIJIYlrrCOYl,LIa
na i B KOMrrJIeKCHOM CJIY'Iae (aa i, j, k - B KBaTepHYlOHHOM).
Tenept, 51 Mory onacart, BTOPOM npmeep y,LIYlBYlT6JIbHOM KBaTepHYlOHYl
3aIJ;YlYl.
IIpHMep 2. OTTMKYlBaHYle ClJIeKTpOHHbIX YPoBHeM, KBaHToBbIM Cl¢
¢eKT XOJIJIa Yl xaparcrepacrn-recrore 'IYlCJIa.
B BeII:\eCTBeHHOM cay-rae pe3YJIbTaT Y2Ke coaepmeaao HeTpYlBYlMeH
Yl 6bIJI ofiaapyzsea, HeCMOTpJl na CBOIO MaTeMaTYl'IecKyIO ¢YH,LIaMeHTMb
HOCTb, JIYlIIlb BCJIe,LICTBYle Pa3BYlTYlJl KBaHTOBOM MeXaHYlKYl (r-ne OH Ha3bIBa
eTCJI TeOpYleM ¢OH HeMMaHa-BYlrHepa). PaCCMOTpYlM MHOroo6pa3Yle acex
ClJIJIYlrrCOB (c nerrrpoi« B Ha'IMe KOOp,LIYlHaT) na eBKJIYl,LIOBOM rrJIOCKOCTYl (YlJIYl,
eCJIYl yrO,LIHO, MHOroo6pa3Yle KBa,LIpaTYl'IHbIX ¢OPM, xo'ropsre YlX 3a,LIaIOT).
* B. N. ApnO.f1,bO, PO,l1CTBeHHMKM c!>aKTopa KOMIIJIeKCHOM IIpoeKTMBHoM IIJIOCKOCTM IIO
KOMIIJIeKCHoMy COIIpJDKeHMIO. Tpyma MaTeM. HHcTMTyTa MM. B. A. CTeKJIOBa PAR,
TOM 224 (1999).
32
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Hexo-ropsre Yl3 ClJIJIYlrrCOB JlBJIJlIOTCJI OKPY2KHOCTJlMYl. Ha rrepBhIM B3rJIJI,LI
Ka2KeTCJI, 'ITO yCJIOBYle 06paIIJ;eHYlJl ClJIJIYlrrCa B OKPY2KHOCTb - ClTO O,LIHO COOT
HOIIleHYle paBeHCTBa no.nyoceii a b, TaK 'ITO rrO,LIMHOroo6pa3Yle OKpy2KHO
CTeM ,LIOJI2KHO YlMeTb KOpa3MepHOCTb O,LIYlH B MHOroo6pa3YlYl acex ClJIJIYlrrCOB.
dTO, O,LIHaKO, He TaK: MHOroo6pa3Yle KBa,LIpaTYl'IHhIX ¢OPM
YlMeeT pa3MepHOCTb 3 (Yl KOOp,LIYlHaThI A, B, 0), a MHOroo6pa3Yle OKpy2KHO
cTeM - paaseepaocr-s 1 (TaK KaK OKPY2KHOCTb C IJ;eHTpOM B HyJIe orrpe,LI6JIJI
eTCJI CBOYlM pa,LIYlYCOM).
BhI,LI6JIJlIOIIJ;ee OKPY2KHOCTYl yCJIOBYle «,LIYlCKpYllvIYlHaHT panen HyJIIO» ,LIJIJI
COOTBeTcTByIOIIJ;erO KBa,LIpaTHOro ypaaaenaa, orrpe,LI6JIJlIOIIJ;erO ,LIJIYlHhI nony-
oceM ClJIJIYlrrCa, YlMeeT BYl,LI (A + 0)2 4(AO - B 2), T. e. CBO,LIYlTCJI K cyMMe
,LIBYX KBa,LIpaTOB, (A - 0)2 + 4B2 0, Yl onpenenser O,LIHOMepHoe rrO,LIMHO-
r-ooopaaae B TpexMepHoM rrpOCTpaHCTBe ¢OPM (a YlMeHHO, rrpmeyto A 0,B 0).
Teopesra Bar-aepa-rpon HeMlVIaHa yTBep2K,LIaeT, 'ITO Yl ,LIJIJI ClJIJIYlrrCOYl
,LIOBB n-MepHOM rrpOCTpaHCTBe npa JII060M n nOd.MiliOZoo6pa3ue 3Jl,Jl,uncoudo66paw,e'liUJ! UMeem xopoauepnocm« dea, TaK 'ITO 'liU dJl,J! 3J1,JI,uncouda otnueeonOJl,O;)ICe'liuJ!, 'liU dJl,J! 'lMe'li06 Od'liOnapaMempU"iec",;ozo ceueticmea otnueeo noJl,o;)ICe'liuJ! 3J1,JI,uncoudbl 6paw,e'liUJ! 'lie ecmpewasomcs:
ECJIYl HapYlCOBaTb r-parpax 3aBYlCYlMOCTYl ,LIJIYlH n no.nyoceii am (P) ,LIJIJI
ClJIJIYlrrCOYl,LIa TaKOrO ceMeMCTBa, 3aBYlCJlIIJ;erO OT napasre'rpa p, TO na rrJIOC
KOCTYl (p,a) rrOJIY'IaIOTCJI n KpYlBhIX (m 1, ... , n), Ka2K,LIaJI Yl3 KOTOphIX
O,LIH03Ha'IHO npoexr-apyercs na OCb 3Ha'IeHYlM napasee'rpa p, Yl xo'ropsre no
napao He nepecexarorca, XOTJI Yl MoryT YlHOr,LIa rrO,LIXO,LIYlTb ,LIOBOJIbHO 6JIYl3KO
,LIpyr K ,LIpyry (pac. 18).B ¢Yl3YlKe 3Ha'IeHYlJl an Ha3hIBaIOTCJI «YPOBHJlMYl», a Cl¢¢eKT YlX necoana
,LIeHYlJl YlCTOJIKOBhIBaeTCJI KaK «OTTaJIKYlBaHYle» YPoBHeM ,LIpyr ,LIpyroM npa YlX
C6JIYl2KeHYlYl BCJIe,LICTBYle Yl3MeHeHYlJl napasre'rpa.Bnpo-reu, rrOCKOJIbKY ClTa TeopeMa - lVIaTeMaTYl'IeCKaJI, OHa YlMeeT MHO
2KeCTBO pa3HhIX ¢Yl3Yl'IeCKYlX (Yl YlHhIX) rrpYlJI02KeHYlM. Hanpmrep, na BpaIIJ;e
nae crryrtraxa aoxpyr ero IJ;eHTpa Macc CYlJIbHO BJIYlJleT ero «ClJIJIYlrrCOYl,LI naep
IJ;YlYl», Yl, eCJIYl ClTOT ClJIJIYlrrCOYl,LI OKa2KeTCJI ClJIJIYlrrCOYl,LIOM BpaIIJ;eHYlJl, TO yrrpa
BJIJlTb OpYleHTaIJ;YleM Yl KyBhIpKaHYleM TaKOrO crry-maxa JIer'Ie. 'Ieopesra Barnepa-cpon HeMMaHa noxaasraaer , 'ITO dJl,J! mozo, wmotiu cdenanu. 3J1,JI,uncoudu'liepv,uu cnym'liu",;a 3J1,JI,uncoudoM 6paw,e'liuJ!, nedocmamowno nepedeueanu.6doJl,b mma'lizu Od'liy (I'IOCmUp060"i'liY'lO scaccs)»: nodo6'1iblX uuruine doJl,;)IC'lio6blmb 'lie Me'libme d6YX.
5-KOMII. 'f. 33
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PMC. 18. PaCTaJIKMBaHMe cooc-rnennsrx 'fMCeJI.
06paTYlMCJI Terrepb K KOlVIIIJIeKCYl¢YlKaIJ;YlYl TeopeMbI 06 OTTaJIKYlBaHYlYl
YPoBHeM. Kopaaaepaocrt, ,LIBa MHOro06pa3YlJl ClJIJIYlrrCOYl,LIOB BpaIIJ;eHYlJl cpenn
acex ClJIJIYlrrCOYl,LIOB 3aMeHJleTCJI npa nepexo.ae OT KBa,LIpaTYl'IHbIX ¢OPM B JRnK ClpMYlTOBbIM B en na BeIIJ;eCTBeHHyIO KOpa3MepHOCTb, paBHyIO TpeM.
3JIJIYlrrCOYl,LIbI BpaIIJ;eHYlJl He BCTpe'IaIOTCJI 3,L1eCb He TOJIbKO B ornrorrapa
MeTpYl'IeCKYlX ceMeMcTBax 06IIJ;ero rrOJI02KeHYlJl, HO Yl B ,LIByrrapaMeTpYl'IeCKYlX
(a npa Tpex BeIIJ;eCTBeHHbIX napaseerpax ClpMYlTOBbI ClJIJIYlrrCOYl,LIbI C ,LIorrOJI
HYlTeJIbHOM CYlMMeTpYleM BCTpe'IaIOTCJI ,LIJIJI OT,LIeJIbHbIX TO'IeK B TpexMepHoM
rrpOCTpaHCTBe napaxre-rpoa}.H rrO,LIp06HO YlCCJIe,LIOBaJI B03HYlKaIOIIJ;Yle 3,L1eCb TOrrOJIOrYl'IeCKYle sonpocsr
(crpoeaae paCCJIOeHYlJl «c06cTBeHHbIx BeKTopoB», xo'ropsre B MaTeMaTYlKe
cooTBeTcTByIOT rJIaBHbIM OCJlM ClJIJIYlrrCOYl,LIa, a B ¢Yl3YlKe Ha3bIBaIOTCJI «MO
,LIaMYl») B CTaTbe: B. Jil. AP'liOJ/,bd, MO,LIbI Yl KBa3YlMO,LIbI, <f:>yHKIJ;. aHaJIYl3 Yl ero
npaa., T. 6 N~2 (1972), 12-20.CerO,LIHJI ClTYl pmYJIbTaTbI Ha3bIBaIOTCJI TeOpYleM «IJ;eJIO'IYlCJIeHHOrO KBaH
TOBOrO CI¢¢eKTa XOJIJIa» (TaK KaK B03M02KHO ClKCrrepYlMeHTaJIbHOe Ha6JIIO
,LIeHYle rrpOX02K,LIeHYlJl rroBepXHOCTYl B TpexMepHoM rrpocTpaHcTBe-BpeMeHYl,
TO'IKa KOTOPOM orrpe,LIeJIJleTCJI ,LIByMJI rrapaMeTpaMYl, 'Iepe3 CrreIJ;YlaJIbHbIe
TO'IKYl, r,LIe cooTBeTcTBYIOIIJ;Yle ClJIJIYlrrCOYl,LIbI YlMeIOT ,LIOrrOJIHYlTeJIbHYIO CYlM
MeTpYlIO.
B TOrrOJIOrYl'IeCKYlX TepMYlHaX ClTO JlBJIeHYle Ha3bIBaeTCJI «npapameaaesexapaKTepYlCTYl'IeCKOrO 'IYlCJIa ~epHa KOMrrJIeKCHOrO paCCJIOeHYlJl c06cTBeH
HbIX sex'ropoa Ha,LI YKa3aHHOM noaepxaocrtaor npa rrpOX02K,LIeHYlYl CITOM no
BepXHOCTYl -repea CrreIJ;YlaJIbHbIe TO'IKYl. TaKYlM 06pa30M, TOrrOJIOrYl'IeCKaJI Te
0PYlJl IJ;eJIO'IYlCJIeHHOrO KBaHTOBOrO CI¢¢eKTa XOJIJIa 6bIJIa nocrpoena Yl ony
6JIYlKOBaHa B 1972 r. ,LIO ¢Yl3Yl'IeCKOM - rrpOCTO rryTeM KOMrrJIeKCYl¢YlKaIJ;YlYl
KJIaCCYl'IeCKOM TeopeMbI Bar-aepa-cpon HeMMaHa, He,LIOCTaBaJIO TOJIbKO ¢Yl3Yl
'IeCKOM TepMYlHOJIOrYlYl, rrOJlBYlBIII6MCJI rr032Ke.
CerO,LIHJI B TaKOM 2Ke rrOJI02KeHYlYl, KaK 6bIJIa B 1972 r. KOMrrJIeKCHaJI aep
CYlJl, HaxO,LIYlTCJI KBaTepHYlOHHaJI sepcas TeopeMbI Bar-aepa-cpon HeMMaHa.
B CITOM CJIy'Iae BeIIJ;eCTBeHHaJI KOpa3MepHOCTb rrOJlBJIeHYlJl ,LIOrrOJIHYlTeJIb-
34
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HOM CYlMMeTpYlYl (T. e. CTOJIKHOBeHYlJl coficrneamax YPOBHeM rYlrrep3pMYlTO
BhIX ¢OpM) panna na'rn. 3TYl 'IYlCJIa - 2,3 Yl 5 - Y2Ke BCTpe'IaJIYlCb HaM npa
aHaJIYl3e 'IYlCM pe6ep TeTpaCl,LIpa, OKTaCl,LIpa Yl YlKOCaCl,LIpa: Ka2K,LIOe Yl3 HYlX
paBHO d + 1, r,LIe d - BeIIJ;eCTBeHHaJI paaseepaocrt, (BeIIJ;eCTBeHHoM, KOM
rrJIeKCHOM Yl COOTBeTCTBeHHO KBaTepHYlOHHOM rrpJlMoM).
POJIb paCCJIOeHYlJl Xompa 8 3 ---+ 8 2 (CO CJIOeM 8 1), ¢YH,LIaMeHTaJIbHOrO
,LIJIJI KOMrrJIeKCHOM TeOpYlYl KBaHTOBOrO 3¢¢eKTa XOJIJIa, nr-paer B KBaTepHYl
OHHOM cay-rae «aropoe paCCJIOeHYle Xompa», 8 7 ---+ 84 (CO CJIOeM 8 3 ) .
06a 3TYl paCCJIOeHYlJl OrrYlChIBaIOT npoc'ro CTaH,LIapTHOe nocrpoeaae rrpo
eKTYlBHOrO rrpOCTpaHCTBa (CCP1 ~ 8 2 , lHIp1 ~ 84) Yl3 crpepsr cooTBeTcTByIO
mero npoex'raaaouy aercropaor-o npocrpaacraa (83 C CC3
\ 0,87 C lHI2
\ 0):
TO'IKe crpepsr paCCJIOeHYleM COrrOCTaBJIJleTCJI npsxeaa, COe,LIYlHJlIOIIJ;aJI ee C Hy
JIelVI. If, HaKOHeIJ;, «xapaKTepYlCTYl'IeCKYlM KJIaCCaM Yl 'IYlCJIaM L[epHa» KOM
rrJIeKCHOrO CJIy'IaJI (xo-ropsae J1BJIJlIOTCJI KOMrrJIeKCYl¢YlKaIJ;YleM «xapax'repncr-a'IeCKYlX KJIaCCOB illTYl¢MJI-YYlTHYl» Yl «3MJIepOBOM xaparcrepacrmcar BeIIJ;e
CTBeHHOrO CJIy'IaJI) cooTBeTcTByIOT B KBaTepHYlOHHOM cay-rae «xaparcrepnCTYl'IeCKYle KJIaCChI Yl 'IYlCJIa IIoHTpJlrYlHa» (rrepeCTpaYlBaIOIIJ;YleCJI npa rrpo
X02K,LIeHYlYl rrO,LIBYl2KHOrO 'IeThIpeXMepHoro npocrpaacraa napasre'rpoa -repea
CrreIJ;YlaJIbHhIe TO'IKYl rrJlTYlMepHOro rrpOCTpaHCTBa).
Be,LIa JIYlIIlb B TOM, 'ITO ¢Yl3Yl'IeCKYlX HaClBaHYlM 3TYl rOTOBhIe MaTeMaTYl
'IeCKYle pmYJIbTaThI noxa eIIJ;e He rrOJIy'IYlJIYl (XOTJI 51 Yl Ha,LIeIOCb, 'ITO 06Yl
JIYle Pa3JIYl'IHhIX rrOJIeM Yl 'IaCTYlIJ; B cOBpeMeHHoM ¢Yl3YlKe rr03BOJIJlT YKa3aTb Yl
Te 3KCrrepYlMeHTaJIbHO Ha6JIIO,LIaeMhIe CYlTyaIJ;YlYl, xo'ropsre yrrpaBJIJlIOTCJI ona
CaHHOM BhIIIle KBaTepHYlOHHOM TeOpYleM rYlrrep3pMYlTOBhIX MaTpYlIJ;). TPOMKYl
«lR - CC -lHI» Yl «KJIaCChI illTYl¢MJI-YYlTHYl - KJIaCChI L[epHa - KJIaCChI IIoH
TpJlrYlHa» yKaClhIBaIOT, 'ITO KOMrrJIeKCYl¢YlKaIJ;YleM ,LIByx3JIeMeHTHOM r-pyrmsr
Z2 K03¢¢YlIJ;YleHTOB KJIaCCOB illTYl¢MJI-YYlTHYl J1BJIJleTCJI rpyrrrra IJ;MhIX 'IYl
CM Z K03¢¢YlIJ;YleHTOB KJIaCCOB L[epHa, HO 'ITO OKa2KeTCJI KBaTepHYlOHHOM
sepcaeii r-pyrmsr Z2 (YlJIYl KOMrrJIeKCHOM sepcaeii r-pyrmsr IJ;MhIX 'IYlCM Z)noxa eIIJ;e He pemeno (B Ka'IeCTBe KaH,LIYl,LIaTOB BhIcTyrraIOT, B 'IaCTHOCTYl,
Z Yl rpyrrrra Z + iZ IJ;MhIX KOMrrJIeKCHhIX 'IYlCM).
CIII'IHhl 1'1 Koehl
II:JIJI 0603Ha'ieHYlJl ,LIByJIYlCTHOrO HaKphITYlJl r-pyrmsr BpaIIJ;eHYlM 80(3)
r-pynnoii CrrYlHOB Spirr(3) 8 3 8U (2) ¢Yl3YlKYl rrpYl,LIyMaJIYl KpaCYlBhIM Me
TO,LI, oCHoBaHHhIM na MaTeMaTYl'IeCKOM meopuu 'KOC. Ilycr-s na rroBepXHOCTYl
M 3a,LIaHhI n nonapao pa3JIYl'IHhIX TO'IeK. K ocoti U3 n 'liumeu na noeeptmocmu M HaClhIBaeTCJI rrYTb B npocrpaacrne TaKYlX naoopon, Ha'IYlHaIOIIJ;YlMCJI
Yl KOH'IaIOIIJ;YlMCJI B 3a,LIaHHOM HeyrropJl,LIO'IeHHOM naoope n TO'IeK Yl3 M.
5* 35
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IIpYl ClTOM KOCa paCCMaTpYlBaeTCJI KaK rrYTb C TO'IHOCTbIO ,LIO rOMOTOrrYlYl,
T. e. nenpepsraaas ,LIe¢OpMaLJ;YlJl, COXPaHJlIOIIJ;aJI ero Ha'Ia.JIbHhlM Yl KOHe'IHhlM
naoopsr Yl, KOHe'IHO, ,LIe¢OpMYlpYIOIIJ;aJI rrpOMe2KyTO'IHhle Ha60phl TaK, 'IT06hl
OHYl Bce BpeMJI COCTOJlJIYl Yl3 n Pa3JIYl'IHhlX TO'IeK, C'IYlTaeTCJI He MeHJlIOIIJ;eM
KOCy.
H3 KOC Yl3 n HYlTeM M02KHO rrOCTpOYlTb rpyrrrry, rrpo,LIOJI2KaJI O,LIYlH rrYTb
,LIpyrYlM (06paTHaJI xoca - ClTO rryTb, rrpOM,LIeHHhlM B 06paTHOM nanpaaneHYlYl).
Hanpmrep, KOCOM a Yl3 ,LIBYX HYlTeM na OpYleHTYlpyelVIOM rrJIOCKOCTYl JlBJIJI
eTCJI ,LIBYl2KeHYle ClTYlX TO'IeK na rrOJI060pOTa B rrOJI02KYlTMbHYIO CTOpOHy BO
xpyr- cepemmsr COe,LIYlHJlIOIIJ;erO YlX OTpe3Ka. Bce KOChl ak (k ±1, ±2, ... )pa3JIYl'IHhl, HeTpYlBYla.JIbHhl (OTJIYl'IHhl OT HerrO,LIBYl2KHOrO rrYTYl aO e) Yl BMe
CTe C e 06Pa3yIOT (Yl30MOP¢HYIO rpyrrrre IJ;MbIX 'IYlCM Z) rpyrrrry B(2) acex
KOC Yl3 2 HYlTeM na rrJIOCKOCTYl.
Tpyrma B(3) acex KOC na rrJIOCKOCTYl Yl3 Tpex HYlTeM rrOp02K,LIeHa ,LIByMJI
06Pa3YIOIIJ;YlMYl a Yl b, BpaIIJ;aIOIIJ;YlMYl rrapy TO'IeK I, II Yl rrapy TO'IeK II, IIInaoopa JIe2KaIIJ;YlX (B nOM nopamce) na rrpJlMOM TO'IeK (I, II, III), PYlC. 19. 3TYl
,LIBe 06pa3YIOIIJ;Yle y,LIOBJIeTBOpJlIOT COOTHOIIleHYlIO aba bob (,LIOKa3aHHoMY
na PYlC. 19), a ,LIpyrYlX CooTHOIIleHYlM HeT (no ,LIOKa3aTb TpY,LIHee).
PMC. 19. KOChl M3 Tpex HMTeM na IIJIOCKOCTM.
Kocsr na rrJIOCKOCTYl y,LI06HO Yl306pa2KaTb KpYlBhlMYl B TpexMepHOM rrpoc
TpaHCTBe-BpeMeHYl: ,LIJIJI KOChl Yl3 n HYlTeM r-parpax ,LIBYl2KeHYlJl TO'IeK naoopaCOCTOYlT Yl3 n xpaasrx (xo-ropsae Yl Ha3hlBaIOTCJI HYlTJlMYl), rrpYl'IeM, no Tpa
,LIYlIJ;YlYl, OCb aricnacc pacnoaar-aercs BepTYlKa.JIbHO Yl OpYleHTYlpOBaHa caepxy
BHYl3 (ara Tpa,LIYlIJ;YlJl 06'hJlCHJleTCJI 06hl'IHhlM pacrrOJI02KeHYleM 3arrJIeTaeMOM
cnepxy BHYl3 KOChl).
H,LI6JI 06'hJlCHeHYlJl ,LIBYJIYlCTHOrO HaKPhlTYlJl BpaIIJ;eHYlM CrrYlHaMYl COCTOYlT
B TOM, 'IT06hl paCCMOTpeTb KOChl Yl3 n HYlTeM na ccpepe 8 2 (aanpmcep,npa n 4).
36
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B rpyrrrre KOC na rrJIOCKOCTYl HeT ClJIeMeHTOB KOHe'IHOrO nopaaxa: eCJIYl
m-JI creneas KaKOM-JIYl60 KOCbI TPYlBYlaJIbHa, TO OHa Yl cava TPYlBYlaJIbHa. II:JIJI
KOC Yl3 ,LIBYX HYlTeM ClTO O'ieBYl,LIHO, HO Y2Ke npa Tpex HYlTJlX ,LIOKa3aTeJIbCTBO
He CTOJIb rrpOCTO.
Y,LIYlBYlTeJIbHbIM MaTeMaTYl'IeCKYlM ¢aKTOM TeOpYlYl crpepa-recxnx KOC
JlBJIJleTCJI TO, 'ITO B rpyrrrre KOC na crpepe 8 2 6bIBaIOT ClJIeMeHTbI KOHe'IHOrO
nopsmca (aanpauep, BToporo - ,LIJIJI KOC Yl3 'IeTbIpeX HYlTeM).
dTOT TOrrOJIOrYl'IeCKYlM pmyJIbTaT O'IeHb 6JIYl30K K TOMy, 'ITO ¢yH,LIaMeH
TaJIbHaJI rpyrrrra MHoroo6pa3YlJl 80(3) ~ lRp3 COCTOYlT Yl3 ,LIBYX ClJIeMeHTOB,
TaK 'ITO CrrYlHOBOe narcpsrrae 83 ---+ 80(3), COrrOCTaBJIJlIOII:\ee KBaTepHYlOHy
e,LIYlHYl'IHOM HOpMbI 3a,LIaHHOe YlM apameaae, ,LIBYJIYlCTHO.
II. II:YlpaK, KOTOpbIM Yl306p6JI ClTOT MeTO,LI 06'bJlCHeHYlJl CrrYlHOB, npo.ae
MOHCTpYlpOBaJI ¢Yl3YlKaM ClKCrrepYlMeHTaJIbHOe ,LIOKa3aTeJIbCTBO C¢OPMYJIYlPO
BaHHoM TeopeMbI 0 crpepa-recxax KOCax. II:JIJI ClTOrO OH peaJIYl30BaJI Hy2KHyIO
c¢epYl'IecKYIO KOCy Yl3 'IeTbIpeX HYlTeM, HeTpYlBYlaJIbHO CBJl3aB ,LIBe KOHIJ;eH
TpYl'IeCKYle crpepsr 'IeTbIpbMJI BepeBKaMYl B OrpaHYl'IeHHOM YlMYl CJIoe (ClTOT
CJIOM 3aMeHJleT B c¢epYl'IeCKOM CJIy'Iae TpexMepHoe rrpOCTpaHCTBO-BpeMJI,
B KOTOpOM JIe2KaJIYl HYlTYl KOCbI, cooTBeTcTBYIOII:\eM ,LIBYl2KeHYlIO naoopa n TO
'IeK na rrJIOCKOCTYl).
3aTeM BHYTpYl mapa, OrpaHYl'IeHHOrO MeHbII1eM c¢epoM, rrOMeII:\aeTCJI
eme MeHbII1aJI crpepa, COe,LIYlHeHHaJI C YlCXO,LIHO MeHbII1eM (Terrepb Cpe,LIHeM)
c¢epoM TaK 2Ke, KaK Ta 6bIJIa COe,LIYlHeHa C caMoM 60JIbII10M c¢epoM.
If, HaKOHeIJ;, Cpe,LIHJIJI crpepa YHYl'IT02KaeTCJI. IIocJIe ClTOrO 60JIbII1aJI Yl ca
MaJI MeHbII1aJI crpepa OKa3bIBaIOTCJI CBJl3aHHbIMYl 4 BepeBKaMYl TPYlBYlaJIbHbIM
06pa30M (nocae ,LIe¢OpMaIJ;YlYl nepeaxa CTaHOBJlTCJI pa,LIYlaJIbHbIMYl), XOTJI YlC
XO,LIHOe COe,LIYlHeHYle 6bIJIO COBepII1eHHO HeTpYlBYlaJIbHbIM Yl He MOrJIO 6bITb
pacny-raao (C,LIeJIaHO pa,LIYlaJIbHbIM rrOCpe,LICTBOM ,LIe¢OpMaIJ;YlYl B CJIoe Me2K,LIy
OrpaHYl'IYlBaIOII:\YlMYl ero KOHIJ;eHTpYl'IeCKYlMYl C¢epaMYl).
B OTJIYl'IYle OT ¢Yl3YlKOB, MaTeMaTYlKYl ClTOM TeopeMbI TeOpYlYl crpepa-re
CKYlX KOC 06bI'IHO He 3HaIOT, TaK KaK OHYl cnaaasor He YlHTepeCYIOTCJI.
,.lJ;o6aBJIeHHe
OIIpe~eJIeHHe. <f:>yHKIJ;YlJl, rrepeBO,LIJlII:\aJI BCJlKOe HeHyJIeBOe KOMrrJIeKC
Hoe 'IYlCJIO Z B
F(z)
Ha3bIBaeTCJI ¢Y'Ii'K'U,ueu JKy'K06C'KOZO.Teopexra 1. etJy'li'K'U,uJ! JKy'K06C'KOZO nepeeodum O'Kpy'JIC'liocmb Izl r
'lia ruiocxocrm: 'KOMnJl,e'KC'liOU nepescewnot: z 6 ;:JJl,JI,unc C 'U,e'limpOM 6 'liYJl,e C no-
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Jl,yOCJ!MU a, b, zde 2a r + r-1 , 2b r - r-1 , 'lia noocxocmu '1WMnJl,e'KC'liOUnepeMe'li'liOU w F (z) .
,ZJ;OKa3aTeJIhCTBO. BeII:\eCTBeHHaH Yl MHYlMaH 'IaCTYl 'IYlCJIa Z paaasr
r cos 'P Yl r sin 'P, nOClToMY
oTKy,LIa
1
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1- ( cos 'P - i sin 'P),r
'ITO Yl yTBepLK,LIaJIOCb. D<f:>YHKIJ;RlI )[(yKoBcKoro ar-paer or-posrayto POJIb B TeXHYlKe, Yl )[(yKOB
CKYlM BBeJI ee pa,LIYl BhI'IYlCJIeHYlJl nO,LI'helVIHOM CYlJIhI KphIJIa CaMOJIeTa (r-ne OCHO
BaHHaH na HeM ¢opMyJIa )[(yKoBcKoro oCTaeTCJI ¢YH,LIaMeHTaJIbHoM).
Ho OHa YlMeeT Yl MHOrO ,LIpyrYlX npYlJIOLKeHYlM.
Teopexra 2 (Boznraa}, Omotipaucensie eoeeedeuu» 'KOMnJl,e'KC'libIX wu-ceJl, 6 »eaopcm, CC ---+ CC, nepeeodsusee Z 6 W Z2, nepeeoinun Zy'K06 ;:JJl,JI,uncC v,e'limpOM 6 'liYJl,e 6 'libWmO'li06 ;:JJl,JI,unc C ¢O'KyCOM 6 'liYJl,e.
,ZJ;OKa3aTeJIhCTBO. IIYCTb KOMnJIeKCHOe 'IYlCJIO u npooer-aer OKpyLK
HOCTb lui r . Tor-na KOMnJIeKCHOe 'IYlCJIO Z U + u-1 nporier-ae-r (no
TeopeMe 1) ryxoa ClJIJIYlnC C no.ayocmor 2a Yl 2b (c IJ;eHTpOM B HyJIe). 3TYlM
cnocofiose MOLKHO nOJIy'IYlTb ClJIJIYlnC C JII06hIM OTHOIIleHYleM ,LIJIYlH no.nyoceii.
KBa,LIpaT 'IYlCJIa Z eCTb
(1)
Ilepnsre ,LIBa CJIaraeMhIX B cyMMe OnYlChIBaIOT (onJlTb no TeopeMe 1)CHOBa ryxoa ClJIJIYlnC. Ero noayoca YlMeIOT ,LIJIYlHhI 2A r2 - r-2 . KBa
,LIpaT paCCTOJlHYlJl OT IJ;eHTpa ClTOrO ClJIJIYlnCa ,LIO ero ¢oKyca BhI'IYlCJIJleTCJI
no TeopeMe Ilarparopa:
TO eCTb paCCTOJlHYle OT IJ;eHTpa ClJIJIYlnCa {u 2 + u-2 } ,LIO ero rpoxyca eCTb
20 2. DCJIe,LIOBaTeJIbHO, nOCJIe yKa3aHHOrO B ¢opMyJIe (1) C,LIBYlra ClJIJIYlnCa na +2
Ha'IaJIO KOOp,LIYlHaT CTaHeT ¢OKyCOM nepeaeceaaoro ClJIJIYlnCa, 'ITO Yl ,LIOKa3hI
aaer TeopeMy.
3aMe~aHHe. 3TOT pe3YJIbTaT MOLKHO nepecpopwyzapoaam, B ¢Yl3Yl'Ie
CKYlX TepMYlHax: TpaeKTOpYlYl rapMOHYl'IeCKOrO KOJIe6aHYlJl no aaxony Tyxa
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aoxpyr Ha'Ia.JIa KOOp,LI¥'lHaT na rrJIOCKOCTYl KOMrrJIeKCHOrO rrepexeemror-o rrepe
XO,LIJIT npa B03Be,LIeHYlYl KOMrrJIeKCHhIX 'IYlCeJI B KBa,LIpaT B TpaeKTOpYlYl ,LIBYl
2KeHYlJl no 3aKOHy BCeMYlpHOrO TJlrOTeHYlJl (YlJIYl no 3aKOHy rrpYlTJl2KeHYlJl Ky
JIOHa) , CYlJIa KOTOpOrO 06paTHO rrporrOPQYlOHa.JIbHa KBa,LIpaTy paCCTOJlHYlJl
,LIO npa'rnr-aaaromer-o nerrrpa. CalVIO ,LIBYl2KeHYle B Hy2KHOe ,LIBYl2KeHYle npa
ClTOM He nepexomrr: CKOpOCTYl rrpOX02K,LIeHYlJl O,LIHOM Op6YlTbI 060YlMYl ,LIBYl2Ke
HYlJlMYl paaasre,3aMe~aHl'le. 1I:0Ka3aHHaJI TeopeMa BOJIYlHa YlMeeT 3aMe'IaTeJIbHOe 06
06II:\eHYle na CJIy'IaM, KOr,LIa B03Be,LIeHYle B KBa,LIpaT 3aMeHeHO na B03Be,LIeHYle
B ,LIpyryIO crenem, 0:. B ClTOM CJIy'Iae Op6YlTbI ,LIBYl2KeHYlJl B none rrpYlTJl2KeHYlJl
(YlJIYl OTTa.JIKYlBaHYlJl) creneaa A nepexonar B Op6YlTbI ,LIJIJI rrOJIJI c'reneaa B,r,LIe 'IYlCJIa A Yl B CBJl3aHbI COOTHOIIleHYleM ,LIBOMCTBeHHOCTYl,
(A+3)(B+3) 4.
Hanpmrep, 3aKOH Tyxa, ,LIJIJI xo'roporo AI, JlBJIJleTCJI ,LIBOMCTBeHHbIM aa
KOHy BCeMYlpHOrO TJlrOTeHYlJl, ,LIJIJI xo'ropor-o B-2. B Ka'IeCTBe noxaaa
TeJIJI creneaa 0: Hy2KHO B 06II:\eM cay-rae BbI6YlpaTb 3Ha'IeHYle 0: (A + 3)/2(,LIBOMCTBeHHoMy 3aKOHy coo'r-ae-rcr-ayer oripa-raoe npeotipaaoaaaae, 06paT
HOM creneaa (3 (B + 3)/2 1/0:).3a~a~a. HaMTYl aBTO,LIBOMCTBeHHbIe 3aKOHbI. (HbIOTOH YlX Y2Ke YlCCJIe
,LIOBa.JI!)
Y,LIYlBYlTeJIbHbIM 06pa30M, BCJI ara TeOpYlJl ,LIBOMCTBeHHOCTYl rrepeHOCYlTCJI
na CJIy'IaM, KOr,LIa MeCTO CTerreHHOrO npeoopaaoaaaaa W z'" 3aHYlMaeT
JII060M MHOrO'IJIeH (YlJIYl ,LIa2Ke JII06aJI rJIa,LIKaJI B KOMrrJIeKCHOM 06JIaCTYl KOM-
rrJIeKCHaJI ¢YHKQYlJl) W (z) (aarrpauep, W e" YlJIYl W Irr z).II:BoMcTBeHHbIe rrOTeHQYla.JIbI na rrJIOCKOCTJlX {z} Yl {w} ,LIaIOTCJI ¢OPMY
JIaMYl ,LIBYX orrpenenaexcsrx npeoopaaoaanaen BeII:\eCTBeHHbIX ¢YHKQYlM KOM
rrJIeKCHhIX rrepexsenmax,
U(z)
Ilpeotipaaonaaxe nepeecmrr Op6YlTbI ,LIBYl2KeHYlJl TO'IKYl z C rrOCTOJlHHOM
rrOJIHOM ClHeprYleM E B none C rrOTeHQYla.JIbHOM ClHeprYleM U B Op6YlTbI ,LIBYl2Ke
HYlJl TO'IKYl W C rrOCTOJlHHOM rrOJIHOM ClHeprYleM -1/E B none C rrOTeHQYla.JIbHOM
ClHeprYleM V.::::ha y,LIYlBYlTeJIbHaJI ,LIBOMCTBeHHocTb ,LIBYl2KeHYlM B CTOJIb pa3HhIX ,LIByMep
HhIX rrOJIJlX cOXpaHJleTCJI ,LIJIJI KBaHTOBO-MeXaHYl'IeCKOrO ypaaaenas Illpennarepa (R. Faure). Ho HYl KaKYle-JIYl60 0606II:\eHYlJl na ,LIBYl2KeHYlJl B npocrpaa
CTBax ,LIpyroro 'IYlCJIa Yl3MepeHYlM, HYl KBaTepHYlOHHbIe 0606II:\eHYlJl yKa3aHHOM
,LIBOMCTBeHHOCTYl He Yl3BeCTHbI.
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JIHTepaTypa K ~06aBJIeHHIO
1. E. Kasner, Differential-Geometric Aspects of Dynamics, AMS, 19132. T. Needham, Newton and Transmutation of Force, Amer. Math. Monthly,100, N~2 (1993), 119-137.3. B. H. AP'liOJ/,bd, IIceB,LIoKBaTepHMoHHaH reOMeTpMJI, <f>yHKU;. aHa.JIM3 MeronpMJI., N~l (2002).
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