L-13-Spring2018 (Грешнов - Лекции (1-15))

‹…Š–ˆŸ ü13„ â : 08.05.2018®¢¥àå­®á⨠2-£® ¯®à浪 , 㤮¢«¥â¢®àïî騥 ãá«®¢¨î δ 6= 0, ­ §ë¢ îâáï 業âà «ì­ë¬¨. ’®«ìª® ¢ í⮬ á«ãç ¥ á¨á⥬  «¨­¥©­ëå ãà ¢­¥­¨©F(x,y,z)=0,1000 a11 x0 + a12 y0 + a13 z0 + a1F2 (x0 , y0 , z0 ) = 0, ⇔ a21 x0 + a22 y0 + a23 z0 + a2a31 x0 + a32 y0 + a33 z0 + a3F3 (x0 , y0 , z0 ) = 0,= 0,= 0,= 0,(13.1)¯à¨ ¯®¬®é¨ ª®â®à®© ®¯à¥¤¥«ïîâáï 業âàë ¯®¢¥àå­®á⨠, ¨¬¥¥â ¥¤¨­á⢥­­®¥ à¥è¥­¨¥.–¥­âà «ì­ë¥ ¯®¢¥àå­®á⨠¨¬¥îâ ¥¤¨­á⢥­­ë© 業âà ᨬ¬¥âਨ. ®¢¥àå­®áâ¨2-£® ¯®à浪 , ¤«ï ª®â®àëå ¢ë¯®«­ï¥âáï ãá«®¢¨¥ranka11 a21a31a12a22a32a13a23  ≤ 2,a33­ §ë¢ îâáï ­¥æ¥­âà «ì­ë¬¨.

“ â ª¨å ¯®¢¥àå­®á⥩ ¬®¦¥â ¨ ­¥ ¡ëâì 業âà  (á¨á⥬  (13.1) ­¥á®¢¬¥áâ­ ), «¨¡® á¨á⥬  (13.1) ᮢ¬¥áâ­ , ¨ ⮣¤  â®çª¨, ïî騥áï ¥¥ à¥è¥­¨ï¬¨, § ¯®«­ïîâ 楫ãî ¯àï¬ãî (rank=2) ¨«¨ 楫ãî ¯«®áª®áâì(rank=1).«®áª®áâ¨, ᮯà殮­­ë¥ ­ ¯à ¢«¥­¨î.“à ¢­¥­¨¥ ¯«®áª®á⨠, ᮯà殮­­®© ­¥ á¨¬¯â®â¨ç¥áª®¬ã ­ ¯à ¢«¥­¨î, ¬®¦¥â¨¬¥âì á¬ëá« ¨ ¤«ï  á¨¬¯â®â¨ç¥áª®£® ­ ¯à ¢«¥­¨ï (α, β, γ ). ˆ ¢ í⮬ á«ãç ¥ ¯«®áª®áâì π ¬ë ¡ã¤¥¬ ­ §ë¢ âì ¯«®áª®áâìî, ᮯà殮­­®© ­ ¯à ¢«¥­¨î (α, β, γ ).Ž¯à¥¤¥«¥­¨¥ 13.1.

 §®¢¥¬ ¯«®áª®áâìLx + M y + N z + P=0(13.2)¤¨ ¬¥âà «ì­®©, ¥á«¨ áãé¥áâ¢ã¥â å®âï ¡ë ®¤­® ­ ¯à ¢«¥­¨¥ (α, β, γ ), ¤«ï ª®â®à®©íâ  ¯«®áª®áâì ¡ã¤¥â ᮯà殮­  ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠¢â®à®£® ¯®à浪  ,â. ¥.L = a11 α + a12 β + a13 γ, M = a α + a β + a γ,212223(13.3)N=aα+aβ+aγ,313233P = a1 α + a2 β + a3 γ.‘¢®©á⢮ 13.1.

‹î¡ ï ¤¨ ¬¥âà «ì­ ï ¯«®áª®áâì, ᮤ¥à¦¨â ¢á¥ 業âàë , ¥á«¨®­¨ ¥áâì.12„®ª § â¥«ìá⢮. ãáâì (x0 , y0 , z0 ) | â®çª , 㤮¢«¥â¢®àïîé ï ãà ¢­¥­¨ï¬ 業âà  (13.1), ¯®¤áâ ¢¨¬ ¥¥ ¢ ¢ëà ¦¥­¨¥ Lx + M y + N z + P , ⮣¤ , ¨á¯®«ì§ãï (13.3),¬ë ¯®«ã稬(a11 α+a12 β +a13 γ )x0 +(a21 α+a22 β +a23 γ )y0 +(a31 α+a32 β +a33 γ )z0 +a1 α+a2 β +a3 γ= α(a11 x0 + a12 y0 + a13 z0 + a1 ) + β (a21 x0 + a22 y0 + a23 z0 + a2 )+ γ (a31 x0 + a32 y0 + a33 z0 + a3 ) = 0.¥‘¢®©á⢮ 13.2.

’®çª M,¯à¨­ ¤«¥¦ é ï ¢á¥¬ ¤¨ ¬¥âà «ì­ë¬ ¯«®áª®áâאָ®¢¥àå­®á⨠, ï¥âáï 業â஬ ¯®¢¥àå­®á⨠.„®ª § â¥«ìá⢮. ’®çª  M ¯à¨­ ¤«¥¦¨â ¢á¥¬ ¤¨ ¬¥âà «ì­ë¬ ¯«®áª®áâï¬ ¯®¢¥àå­®á⨠, â. ¥. ¯® ¬¥­ì襩 ¬¥à¥ ¢á¥¬ ⥬ ¨§ ­¨å, ª®â®àë¥ á®¯à殮­ë ­¥ á¨¬¯â®â¨ç¥áª¨¬ ­ ¯à ¢«¥­¨ï¬ ¯®¢¥àå­®á⨠(¨ á।¨ ­¨å ­¥ ¬¥­¥¥ âà¥å «¨­¥©­® ­¥§ ¢¨á¨¬ëå­ ¯à ¢«¥­¨© (xi , yi , 1), i = 1, 2, 3, ª ª ¬ë ¤®ª § «¨ à ­¥¥), â. ¥.x1 x2x3y1y2y3  1F1 (x0 , y0 , z0 )0F1 (x0 , y0 , z0 )01F2 (x0 , y0 , z0 ) = 0 ⇒ F2 (x0 , y0 , z0 ) = 0  ,1F3 (x0 , y0 , z0 )0F3 (x0 , y0 , z0 )0®âªã¤  á«¥¤ã¥â, á¬. (13.1), çâ® â®çª  M | 業âà ᨬ¬¥âਨ ¯®¢¥àå­®á⨠.‘¢®©á⢮ 13.3. 10 …᫨ ¤«ï  á¨¬¯â®â¨ç¥áª®£® ­ ¯à ¢«¥­¨ï¥= (α, β, γ ) áãé¥áâ¢ã¥â ¤¨ ¬¥âà «ì­ ï ¯«®áª®áâì á ­®à¬ «ì­ë¬ ¢¥ªâ®à®¬ n = (L, M, N ), â®v⊥n (¯«®áª®áâì ¯ à ««¥«ì­  ­ ¯à ¢«¥­¨î); 20 ¥á«¨ ­ ¯à ¢«¥­¨¥ v = (α, β, γ ) ¯ à ««¥«ì­® ᮯà殮­­®© ¥¬ã ¯«®áª®áâ¨ á ­®à¬ «ì­ë¬ ¢¥ªâ®à®¬ n = (L, M, N ),â® ­ ¯à ¢«¥­¨¥ v = (α, β, γ )  á¨¬¯â®â¨ç¥áª®¥.v„®ª § â¥«ìá⢮.

‘¢®©á⢮ 13.3 ¢ë⥪ ¥â ¨§ á«¥¤ãî饣® ⮦¤¥á⢠(αβL(αγ)M  =Nβγ ) a11 a21a31a12a22a32 αa13a23   β  .a33㥑¢®©á⢮ 13.4. ãáâì ¯«®áª®áâìP1 ᮯà殮­  ­ ¯à ¢«¥­¨î v1 = (α1 , β1 , γ1 ),¯«®áª®áâì P2 ᮯà殮­  ­ ¯à ¢«¥­¨î v2 = (α2 , β2 , γ2 ). ’®£¤  ¯«®áª®áâì λP1 + µP2ᮯà殮­  ­ ¯à ¢«¥­¨î λv1 + µv2 , λ, µ ∈ R.„®ª § â¥«ìá⢮ ¢ë⥪ ¥â ¨§ ®¯à¥¤¥«¥­¨ï 13.1.

(à®¢¥àìâ¥!)¥3‘¢®©á⢮ 13.5. ‹î¡ ï ¯«®áª®áâì P1 , ¯à®å®¤ïé ï ç¥à¥§ 業âà M0 業âà «ì­®©¯®¢¥àå­®á⨠, ᮯà殮­  ­¥ª®â®à®¬ã ­ ¯à ¢«¥­¨î (α, β, γ ).„®ª § â¥«ìá⢮. ãáâì (x0 , y0 , z0 ) | ª®®à¤¨­ âë â®çª¨ M0 . “à ¢­¥­¨¥ ¯«®áª®áâ¨P1 , ¯à®å®¤ï饩 ç¥à¥§ â®çªã M0 , ¨¬¥¥â ¢¨¤A(x − x0 ) + B (y − y0 ) + C (z − z0 ) = Au + Bv + Cw= 0,(13.4)£¤¥ A, B, C , A2 + B 2 + C 2 6= 0, | ­¥ª®â®àë¥ ª®­áâ ­âë. ®¯à®¡ã¥¬ § ¯¨á âìãà ¢­¥­¨¥ (13.4) ¤«ï P1 ¢ ¢¨¤¥αF1 (x, y, z ) + βF2 (x, y, z ) + γF3 (x, y, z ) = 0.ˆá¯®«ì§ãï ãà ¢­¥­¨ï, ®¯à¥¤¥«ïî騥 â®â ä ªâ, çâ®¬ë ¯®«ãç ¥¬ ⮦¤¥á⢮M0(13.5)| 業âà ¯®¢¥àå­®á⨠,αF1 (x, y, z ) + βF2 (x, y, z ) + γF3 (x, y, z )= αF1 (x − x0 + x0 , y − y0 + y0 , z − z0 + z0 ) + βF2 (x − x0 + x0 , y − y0 + y0 , z − z0 + z0 )+ γF3 (x − x0 + x0 , y − y0 + y0 , z − z0 + z0 )= α(a11 u + a12 v + a13 w) + β (a21 u + a22 v + a23 w) + γ (a31 u + a32 v + a33 w) = 0.(13.6)ˆ§ (13.4), (13.6) ¢ë⥪ ¥â á«¥¤ãîé ï ­¥¢ë஦¤¥­­ ï (δ 6= 0) á¨á⥬  «¨­¥©­ëåãà ¢­¥­¨©, ®¤­®§­ ç­® ®¯à¥¤¥«ïîé ï ­ ¯à ¢«¥­¨¥ (α, β, γ ):a11 a21a31a12a22a32   a13αAa23β = B .Ca33γ’ ª¨¬ ®¡à §®¬, ¬ë ¤®ª § «¨, çâ® ¯«®áª®áâì P1 ¨¬¥¥â ¢¨¤ (13.5).¥‘®¯à殮­­ë¥ ­ ¯à ¢«¥­¨ï.

Žá®¡ë¥ ­ ¯à ¢«¥­¨ïŽ¯à¥¤¥«¥­¨¥ 13.2.  ¯à ¢«¥­¨ï (α1 , β1 , γ1 ), v2 = (α2 , β2 , γ2 ) ­ §ë¢ îâáï ᮯà殮­­ë¬¨ ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠, ¥á«¨ ¢ë¯®«­¥­® á«¥¤ãî饥 ãá«®¢¨¥( α1β1α2γ1 ) (A )  β  = 0.12γ2¥á«®¦­® ¤®ª § âì (â ª ¦¥, ª ª ¨ ¢ á«ãç ¥ ªà¨¢ëå 2-£® ¯®à浪 ), çâ® ®¯à¥¤¥«¥­¨¥ (13.2) ­¥ § ¢¨á¨â ®â ¢ë¡®à   ä䨭­®© á¨áâ¥¬ë ª®®à¤¨­ â.4“ ¤ ­­®£® ­ ¯à ¢«¥­¨ï (α, β, γ ) ­¥â ᮯà殮­­®© ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠2-£®¯®à浪  ¯«®áª®á⨠Lx + M y + N z + P = 0, ¥á«¨0 = a11 α + a12 β + a13 γ = L,0 = a21 α + a22 β + a23 γ = M,0 = a31 α + a32 β + a33 γ = N.’ ª®¥ ­ ¯à ¢«¥­¨¥ ­ §ë¢ îâ ®á®¡ë¬ ­ ¯à ¢«¥­¨¥¬ ¯®¢¥àå­®á⨠.‘¢®©á⢮ 13.6.

10 Žá®¡®¥ ­ ¯à ¢«¥­¨¥ ᮯà殮­® á ¬®¬ã ᥡ¥ (á ¬®á®¯à殮­­®¥),  ¯®â®¬ã ï¥âáï  á¨¬¯â®â¨ç¥áª¨¬; 20 ç¨á«® «¨­¥©­® ­¥§ ¢¨á¨¬ëå ®á®¡ëå ­ ¯à ¢«¥­¨© ­¥ ¯à¥¢®á室¨â ¢¥«¨ç¨­ë 3 − rank A1 ; 30 Žá®¡®¥ ­ ¯à ¢«¥­¨¥ ­¥ § ¢¨á¨â®â ¢ë¡®à   ä䨭­®© á¨áâ¥¬ë ª®®à¤¨­ â.„®ª § â¥«ìá⢮. . 10 ®ç¥¢¨¤¥­, ¯. 20 ï¥âáï ¯à®áâë¬ á«¥¤á⢨¥¬ ⥮ਨ «¨­¥©­®©  «£¥¡àë.

30 ãáâì u = (a, b, c) | ®á®¡®¥ ­ ¯à ¢«¥­¨¥, â. ¥. A1 u = 0 ∈ R3 . €ä䨭­ ï § ¬¥­  ª®®à¤¨­ â ¨¬¥¥â ¢¨¤ v­ = Cv + c, £¤¥ v­ = (x­ , y­ , z­ ), v = (x, y, z ),det C 6= 0, c ∈ R3 . ’®£¤  ¢ ý­®¢®©þ á¨á⥬¥ ª®®à¤¨­ â ­ ¯à ¢«¥­¨¥ u § ¯¨á뢠¥âá猪ª C −1 u, ¬ âà¨æ  A1 | ª ª C −1 A1 C , ¨ ¬ë ¨¬¥¥¬ C −1 A1 CC −1 u = C −1 A1 u = 0.¥à¥¤«®¦¥­¨¥ 13.1.

…᫨ (α1 , β1 , γ1 ) | ­¥ ®á®¡®¥ ­ ¯à ¢«¥­¨¥, ⮠ᮯà殮­-­ë¥ ¥¬ã ïîâáï ⥠¨ ⮫쪮 â¥, ª®â®àë¥ «¥¦ â ¢ ¯«®áª®áâ¨, ᮯà殮­­®©­ ¯à ¢«¥­¨î (α1 , β1 , γ1 ) ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠.„®ª § â¥«ìá⢮. «®áª®áâì Lx + M y + N z + P = 0 ᮯà殮­  ­ ¯à ¢«¥­¨î(α1 , β1 , γ1 ) ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠⇐⇒LMNP= a11 α1 + a12 β1 + a13 γ1 ,= a21 α1 + a22 β1 + a23 γ1 ,= a31 α1 + a32 β1 + a33 γ1 ,= a1 α1 + a2 β1 + a3 γ1 . a11L⇒ M = a21Na31a12a22a32a13α1a23   β1 a33γ1 .(13.7)‚¥ªâ®à (α, β, γ ) «¥¦¨â ¢ ¯«®áª®á⨠Lx + M y + N z + P = 0⇐⇒ Lα + M β + N ㊮¬¡¨­¨àãï (13.7), (13.8), ¯®«ãç ¥¬ ¯à¥¤«®¦¥­¨¥ 13.1.à¥¤«®¦¥­¨¥ 13.2.  ¯à ¢«¥­¨¥ (λ, µ, ν ) ®á®¡®¥= 0. (13.8)¥¢¥ªâ®à (λ, µ, ν ) ¯ à ««¥«¥­«î¡®© ¤¨ ¬¥âà «ì­®© ¯«®áª®á⨠¯®¢¥àå­®á⨠2-£® ¯®à浪  .„®ª § â¥«ìá⢮. (⇒) ãáâì Lx + M y + N z + P = 0 | ¤¨ ¬¥âà «ì­ ï ¯«®áª®áâì,ᮯà殮­­ ï ­ ¯à ¢«¥­¨î (α, β, γ ).

“á«®¢¨¥ ¯ à ««¥«ì­®á⨠­ ¯à ¢«¥­¨ï ­¥ª®â®~ γ~) á®á⮨⠢ ⮬, çâ® α~ N + βM~ +~γ N = 0. ‘ ¤à㣮© áâ®à®­ë,ண® ­ ¯à ¢«¥­¨ï (~α, β,⇔5~ γ~) | ®á®¡®¥ ­ ¯à ¢«¥­¨¥, ⮥᫨ (~α, β,( α~ β~ γ~ ) (A1 ) = ( 0 0 0 ) ⇒ ( α~ β~ α( α~γ~ ) (A1 ) β  =γLβ~ γ~ ) MN= 0.(⇐) ãáâì (αi , βi , γi ), i = 1, 2, 3, | ­¥ª®¬¯« ­ à­ë¥ ­¥ á¨¬¯â®â¨ç¥áª¨¥ ­ ¯à ¢«¥­¨ï ¯®¢¥àå­®á⨠, áãé¥á⢮¢ ­¨¥ ª®â®àëå ­ ¬¨ à ­¥¥. ’®£¤  ¬ë ¨¬¥¥¬0 = (a11 αi + a12 βi + a13 γi )λ + (a21 αi + a22 βi + a23 γi )µ + (a31 αi + a32 βi + a33 γi )ν= (a11 λ + a12 µ + a13 ν )αi +(a21 λ + a22 µ + a23 ν )βi +(a31 λ + a32 µ + a33 ν )γi ,ˆ§ (13.9) á«¥¤ã¥â, çâ® aj 1 λ + aj 2 µ + aj 3 ν = 0 ¤«ï­ ¯à ¢«¥­¨¥ (λ, µ, ν ) | ®á®¡®¥.à¨¬¥à 13.1.

10 “ ¯®¢¥àå­®á⥩ x2± px2a2 ± x2a2 ±y2qy2b2y2b2= 2z,ji = 1, 2, 3.(13.9)= 1, 2, 3,   íâ® ¨ §­ ç¨â, ç⮥p, q > 0,= ±1,=0¥¤¨­á⢥­­®¥ ®á®¡®¥ ­ ¯à ¢«¥­¨¥ | ¢¥ªâ®à (0, 0, 1).20 “ ¯®¢¥àå­®á⥩½ 2y = 2px,y2 = c®á®¡ë¥ ­ ¯à ¢«¥­¨ï | ¢á¥ ­ ¯à ¢«¥­¨ï, ¯ à ««¥«ì­ë¥ ¯«®áª®á⨠y = 0.ƒ« ¢­ë¥ ­ ¯à ¢«¥­¨ï ¯®¢¥àå­®á⨠2-£® ¯®à浪 Ž¯à¥¤¥«¥­¨¥ 13.3.

 ¯à ¢«¥­¨¥ (α, β, γ ) ­ §ë¢ ¥âáï £« ¢­ë¬ ­ ¯à ¢«¥­¨¥¬ ¯®¢¥àå­®á⨠2-£® ¯®à浪  , ¥á«¨ ®­® ¯¥à¯¥­¤¨ªã«ïà­® ¢á¥¬ ᮯà殮­­ë¬ ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠¥¬ã ­ ¯à ¢«¥­¨ï¬.‹î¡®¥ ®á®¡®¥ ­ ¯à ¢«¥­¨¥ | £« ¢­®¥ (®á®¡®¥ ­ ¯à ¢«¥­¨¥ ᮯà殮­® «î¡®¬ã­ ¯à ¢«¥­¨î, ¢ ⮬ ç¨á«¥ ¨ ¥¬ã ¯¥à¯¥­¤¨ªã«ïà­®¬ã).

…᫨ ¦¥ (α, β, γ ) | £« ¢­®¥,­® ­¥ ®á®¡®¥ ­ ¯à ¢«¥­¨¥, â® ®­® ¯¥à¯¥­¤¨ªã«ïà­® ᮯà殮­­®© ®â­®á¨â¥«ì­® ¯®¢¥àå­®á⨠¥¬ã ¤¨ ¬¥âà «ì­®© ¯«®áª®á⨠Lx + M y + N z + P = 0. ’®£¤ , ¨á¯®«ì§ãﮯ।¥«¥­¨¥ ¤¨ ¬¥âà «ì­®© ¯«®áª®áâ¨, ¬ë ¯®«ãç ¥¬, çâ®LMN= a11 α + a12 β + a13 γ = λα,= a21 α + a22 β + a23 γ = λβ,= a31 α + a32 β + a33 γ = λγ.(13.10)6‚á类¥ £« ¢­®¥ ­ ¯à ¢«¥­¨¥ 㤮¢«¥â¢®àï¥â á¨á⥬¥ ãà ¢­¥­¨© (13.10), ¯à¨ç¥¬¤«ï ®á®¡ëå ­ ¯à ¢«¥­¨© ¬ë ¨¬¥¥¬ λ = 0,   ¤«ï ­¥ ®á®¡ëå λ 6= 0.