L-5-Spring2018 (Грешнов - Лекции (1-15))

‹…Š–ˆŸ ü5„ â : 06.03.2018µa11a21a12a22¶ áᬮâਬ ᮡá⢥­­ë¥ ç¨á«  λ1 , λ2 ¬ âà¨æë A1 =. …᫨ λ1 6= λ2 ,â® ¨§ ¯. 10 «¥¬¬ë 4.1 ¢ë⥪ ¥â, ç⮠ᮡá⢥­­ë¥ ¢¥ªâ®àë ¬ âà¨æë A1 ¨¬¥îâ ¢¨¤(cos α, sin α), (− sin α, cos α) ¤«ï ­¥ª®â®à®£® 㣫  α. ãáâì λ1 = λ2 ; íâ® ¢®§¬®¦­®«¨èì ¢ ⮬ á«ãç ¥, á¬. ä®à¬ã«ã (4.12) ¨§ «¥ªæ¨¨ ü4, ª®£¤  a11 = a22 , a12 = 0.’®£¤  ®ç¥¢¨¤­®, çâ® a11 = λ1 , ¨ α = 0.‚뢥¤¥¬ ã­¨¢¥àá «ì­ãî ä®à¬ã«ã ¤«ï ­ å®¦¤¥­¨ï 㣫  α. „®áâ â®ç­® à áᬮâà¥âì á«ãç © a12 6= 0. ˆ§ ä®à¬ã«ë (4.12) ¨§ «¥ªæ¨¨ ü4 á«¥¤ã¥â, çâ® λ1 6= λ2 . ãáâì(cos α, sin α) | ᮡá⢥­­ë© ¢¥ªâ®à ¬ âà¨æë A1 , ᮮ⢥âáâ¢ãî騩 ᮡá⢥­­®¬ãç¨á«ã λ1 . Œë ¨¬¥¥¬µa11 − λ1a12a12a22 − λ1¶µ¶µ ¶cos α = 0sin α0,á«¥¤®¢ â¥«ì­®,µh(a11 − λ1 , a12 ), (cos α, sin α)i = 0 ⇒®âªã¤  ¢ë⥪ ¥âtg α =¶cos α = ksin αµ−a12a11 − λ1¶,k 6= 0,λ1 − a11.a12(5.1)Žâ¬¥â¨¬, çâ® ¯à¨ λ1 = 0 ä®à¬ã«  (5.1) ¨¬¥¥â ¢¨¤tg α = −a11.a12(5.2)ˆ§ ä®à¬ã«ë (5.1) á«¥¤ã¥â, çâ®λ −aa11112sin α = ± p, cos α = ± p.2(λ1 − a11 )2 + a12(λ1 − a11 )2 + a212(5.3)’ ª¨¬ ®¡à §®¬, à ¢¥­á⢠ (5.3) ¤ îâ ­ ¬ ¤¢  ¢¥ªâ®à .

Ž¡ëç­® ¨§ ­¨å ¢ë¡¨à îâ â®â, ª®â®àë© ®¡à §ã¥â á ®áìî OX ­¥ â㯮© 㣮«. ‚ë¡à ­­ë© â ª¨¬ ®¡à §®¬á®¡á⢥­­ë© ¢¥ªâ®à (cos α, sin α), ᮮ⢥âáâ¢ãî騩 ᮡá⢥­­®¬ã ç¨á«ã λ1 , ¨£à ¥â஫ì ý¯¥à¢®£®þ ¢¥ªâ®à  ¢ ­®¢®© ¯àאַ㣮«ì­®© ª®®à¤¨­ â­®© á¨á⥬¥.„ «ì­¥©è¨¥ ¯à¥®¡à §®¢ ­¨ï, á¢ï§ ­­ë¥ á ãà ¢­¥­¨¥¬λ1 (x­ )2 + λ2 (y­ )2 + 2x­ a01 + 2y­ a02 + a0ᬠ«¥ªæ¨î ü4, à §¡¨¢ îâáï ­  ¤¢  á«ãç ï.1= 0,(5.4)2–¥­âà «ì­ë© á«ãç © áᬮâਬ á«ãç ©, ª®£¤  δ = λ1 λ2 6= 0 (業âà «ì­ë© á«ãç ©). „«ï ⮣®, çâ®-¡ë ¯à¨¢¥á⨠¢ 業âà «ì­®¬ á«ãç ¥ ªà¨¢ãî ª ª ­®­¨ç¥áª®© ä®à¬¥, ¯à®é¥ ¯à¥¦¤¥,祬 ¤¥« âì ¯®¢®à®â ­  㣮« α, ¯¥à¥å®¤ï ¯à¨ í⮬ ®â ®¡é¥£® ãà ¢­¥­¨ï ªà¨¢®© 2-£®¯®à浪 F (x, y ) = a11 x2 + 2a12 xy + a22 y 2 + 2a1 x + 2a2 y + a0 = 0(5.5)ª ãà ¢­¥­¨î (5.4), ᤢ¨­¥¬ ­ ç «® ª®®à¤¨­ â ¢ â®çªã (x0 , y0 ) â ª, çâ®¡ë ¢ãà ¢­¥­¨¨ (5.5) ¨á祧«¨ «¨­¥©­ë¥ ç«¥­ë.

„«ï í⮣® ¢¢®¤¨¬ ­®¢ë¥ ¯¥à¥¬¥­­ë¥½x = x0 + ξ,y = y0 + η,¨ ¤ «¥¥ ¤¥« ¥¬ § ¬¥­ã ¯¥à¥¬¥­­ëå ¢ (5.3):a11 (x0 + ξ )2 + 2a12 (x0 + ξ )(y0 + η ) + a22 (y0 + η )2 + 2a1 (x0 + ξ ) + 2a2 (y0 + η ) + a0= a11 ξ 2 +2a12 ξη + a22 η2 +2ξ (a11 x0 + a12 y0 + a1 )+2η(a12 x0 + a22 y0 + a2 )+ F (x0 , y0 ) = 0.(5.6)®­ïâ­®, çâ®¡ë § ­ã«¨âì «¨­¥©­ë¥ ¢ (5.6) ç«¥­ë, ­¥®¡å®¤¨¬® ­ ©â¨ à¥è¥­¨¥ (x0 , y0 )á¨á⥬뵶µ ¶ µ¶a11 a12x0−a1= −a ;(5.7)aay122202à¥è¥­¨¥ á¨á⥬ë (5.7) áãé¥áâ¢ã¥â ¨ ¥¤¨­á⢥­­®, â ª ª ª δ 6= 0. ®á«¥ ⮣®, ª ª¬ë ­ è«¨ à¥è¥­¨¥ (x0 , y0 ) á¨á⥬ë (5.17) ¨ ¯à®¨§¢¥«¨ § ¬¥­ã ¢ (5.5), ¬ë ¯®«ã稬ãà ¢­¥­¨¥a11 ξ 2 + 2a12 ξη + a22 η 2 + F (x0 , y0 ) = 0.(5.8)„ «¥¥ ¢ë¯®«­ï¥¬ ¯®¢®à®â ª®®à¤¨­ â­ëå ®á¥©µcos αsin α− sin αcos ᶵx­y­¶µ ¶= ηξ¨ ¯à¨å®¤¨¬ ª ãà ¢­¥­¨îλ1 (x­ )2 + λ2 (y­ )2 + F (x0 , y0 ) = 0.(5.9)Žç¥¢¨¤­®, çâ® ­ ç «® ª®®à¤¨­ â (x­ , y­ ) ï¥âáï 業â஬ ᨬ¬¥âਨ ªà¨¢®© γ :¥á«¨ (x­ , y­ ) 㤮¢«¥â¢®àïîâ (5.9), â® ¨ (−x­ , −y­ ) ®ç¥¢¨¤­® 㤮¢«¥â¢®àïîâ (5.9).‘«¥¤®¢ â¥«ì­®, â®çª  (x0 , y0 ) | 業âà ᨬ¬¥âਨ ªà¨¢®©, ®¯à¥¤¥«ï¥¬®© ãà ¢­¥­¨¥¬ (5.5), ¨ íâ®â 業âà ®¯à¥¤¥«ï¥âáï ª ª ¥¤¨­á⢥­­®¥ à¥è¥­¨¥ á¨á⥬ë (5.7)(¢ í⮬ | á¬ëá« ­ §¢ ­¨ï ý業âà «ì­ë© á«ãç ©þ).3’ ª¨¬ ®¡à §®¬, ­ ¬¨ ãáâ ­®¢«¥­  á«¥¤ãîé ï’¥®à¥¬  5.1.

ãáâì ¤ ­  ªà¨¢ ï 2-£® ¯®à浪  γ , ®¯à¥¤¥«ï¥¬ ï ãà ¢­¥­¨¥¬F (x, y ) = a11 x2 + 2a12 xy + a22 y 2 + 2a1 x + 2a2 y + a0 = 0,µ¶a11 a12δ = det6 0. ’®£¤  ¢ ¯àאַ㣮«ì­®© á¨á⥬¥ ª®®à¤¨­ â (x­ , y­ ) á=a12 a22業â஬ ¢ â®çª¥ á ª®®à¤¨­ â ¬¨ (x0 , y0 ), ®¯à¥¤¥«ï¥¬ë¬¨ ª ª à¥è¥­¨¥ ãà ¢­¥­¨©µ¶µ ¶ µ¶a11 a12x0−a1= −a ,a12 a22y02®áì  ¡æ¨áá ª®â®à®© ­ ª«®­¥­  ª ¯¥à¢®­ ç «ì­®© ®á¨  ¡æ¨áá ¯®¤ 㣫®¬ α, ®¯à¥11 , £¤¥ λ | ᮡá⢥­­®¥ ç¨á«® ¬ âà¨æ뤥«ï¥¬ë¬ ¶¨§ ⮦¤¥á⢠ tg α = λ1a−a112µa11a12a12a22, ãà ¢­¥­¨¥ ªà¨¢®© γ ¨¬¥¥â ¢¨¤λ1 (x­ )2 + λ2 (y­ )2 + F (x0 , y0 ) = 0.  ®á­®¢¥ ⥮६ë 5.1 ¯à®¨§¢¥¤¥¬ ª« áá¨ä¨ª æ¨î ªà¨¢ëå 2-£® ¯®à浪  ¢ 業âà «ì­®¬ á«ãç ¥.F (x0 , y0 ) = 0, δ < 0 | ¤¢¥ ¯¥à¥á¥ª î騥áï ¤¥©á⢨⥫ì­ë¥ ¯àï¬ë¥.F (x0 , y0 ) 6= 0, δ < 0 | £¨¯¥à¡®« .F (x0 , y0 ) = 0, δ > 0 | ¤¢¥ ¬­¨¬ë¥ ¯¥à¥á¥ª î騥áï ¯àï¬ë¥.F (x0 , y0 ) 6= 0, δ > 0 (⇒ λ1 , λ2 ®¤­®£® §­ ª ): í««¨¯á (SF (x0 , y0 ) < 0), ¬­¨¬ë©í««¨¯á (SF (x0 , y0 ) > 0). à ¡®«¨ç¥áª¨© á«ãç © áᬮâਬ ¯ à ¡®«¨ç¥áª¨© á«ãç © ªà¨¢ëå 2-£® ¯®à浪 , â.

¥.= 0. ‚í⮬ á«ãç ¥, ãç¨â뢠ï ãá«®¢¨¥ δ 6= 0, ¢á¥£¤  ¢ë¯®«­ï¥âáï a211 + a222 6= 0. ®« £ ¥¬λ1 = 0, ⮣¤  S = λ2 6= 0. ãáâì a12 6= 0 (¢ ¯à®â¨¢­®¬ á«ãç ¥, ãç¨â뢠ï ãá«®¢¨¥δ = 0, ¬ë ¯à¨å®¤¨¬ ª ãà ¢­¥­¨î ⨯  a22 y 2 + 2a1 x + 2a2 y + a0 = 0). ‚¢¥¤¥¬ ­®¢ë¥¯¥à¥¬¥­­ë¥ (ξ, η) ¯® ¯à ¢¨«ãµcos αsin α− sin αcos α䶵 ¶ µ ¶ξ= xy ,η(5.10)£¤¥ 㣮« α ®¯à¥¤¥«ï¥âáï ¨§ (5.7). à¨ ¯®¬®é¨ (5.10) ¯à®¨§¢¥¤¥¬ ¢ (5.5) § ¬¥­ã¯¥à¥¬¥­­ëå, ¢ १ã«ìâ â¥ ¯®«ã稬λ2 η 2 + 2a1 (ξ cos α − η sin α) + 2a2 (ξ sin α + η cos α) + a0= λ2 η2 + 2ξ (a1 cos α + a2 sin α) + 2η(−a1 sin α + a2 cos α) + a0= λ2 η2 + 2a01 ξ + 2a02 η + a0 = 0.(5.11)4ˆ§ã稬 ¯®¤à®¡­¥¥ ãà ¢­¥­¨¥ (5.11).‘«ãç © I.

a01 = a1 cos α + a2 sin α 6= 0. ¥á«®¦­® ¢¨¤¥âì, çâ® ¢ ᨫã (5.2) ¤ ­­®¥ãá«®¢¨¥ ãá«®¢¨¥ íª¢¨¢ «¥­â­® ⮬ã, çâ® a1 a12 − a2 a11 6= 0. ’. ª. a12 6= 0, â®|tg α| < ∞, ¨ ¬ë ¨¬¥¥¬ sin α = ± √ 2a11 2 ¨ cos α = ∓ √ 2a12 2 .)a11 +a12a11 +a12‚ á«ãç ¥ I ãà ¢­¥­¨¥ (5.11) ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥­® ¢ ¢¨¤¥³ a0 ´2 ´³a02a0 λ2 − (a02 )2 ´220= λ2 η + 2 η ++ 2a 1 ξ +λ2λ22a01 λ2³ a 0 ´2³ a λ − (a0 )2 ´= λ2 η+ 2 +2a01 ξ + 0 2 0 2= λ2 (η−η0 )2 +2a01 (ξ−ξ0 ) = λ2 (y­ )2 +2a01 x­λ22a1 λ2³λ2 η 2 + 2a01 ξ + 2a02 η + a0¨«¨ ¦¥y­2=−2a01λ2x­=−2a01S= 0,(5.12)x­ .„«ï ⮣®, çâ®¡ë ¢¥«¨ç¨­  − 2λa21 ¡ë«  ¡ë ¯®«®¦¨â¥«ì­®©, ¬ë ¤®«¦­ë ¢ë¡à â좥ªâ®à (cos α, sin α) â ª¨¬ ®¡à §®¬, ç⮡ë0Sa01= λ2 (a1 cos α + a2 sin α) < 0,  íâ® ¢á¥£¤  ¬®¦­® ᤥ« âì, ¯®áª®«ìªãa11,a211 + a212sin α = ± pa12.a211 + a212cos α = ∓ pŠ®®à¤¨­ âë (x0 , y0 ) 業âà  ª ­®­¨ç¥áª®© á¨áâ¥¬ë ª®®à¤¨­ â ¢ á«ãç ¥ I ý¢ áâ à®©á¨á⥬¥ ª®®à¤¨­ âþ ¢ëç¨á«ïîâáï ¯® ä®à¬ã«¥µcos αsin α− sin αcos ᶵξ0η0¶µ=x0y0¶.“à ¢­¥­¨¥ (5.12) ¯à¥¤áâ ¢«ï¥â ᮡ®© ª ­®­¨ç¥áª®¥ ãà ¢­¥­¨¥ ¯ à ¡®«ë.‘«ãç © II.

a01 = a1 cos α + a2 sin α = 0. ’®£¤  (5.11) ¨¬¥¥â ¢¨¤λ2 η 2 + 2ηa02 + a0= 0.(5.13)à¥®¡à §ã¥¬ ãà ¢­¥­¨¥ (5.13):λ2 η 2 + 2ηa02 + a0ý‘¤¢¨­¥¬þ ª®®à¤¨­ â뵶2a02(a0 )2= λ2 η ++ a0 − 2λ2λ2(= ξ,ay­ = η + λ22 ,x­0= 0.(5.14)5⮣¤  ãà ¢­¥­¨¥ (5.14) ¯à¨¬¥â ¢¨¤a0 λ2 − (a02 )2λ2 y­2 +λ2= 0 ⇐⇒ y­2 +KS2= 0,K= a0 λ2 − (a02 )2 = a0 S − (a02 )2 . (5.15)“à ¢­¥­¨¥ (5.15) ¯à¥¤áâ ¢«ï¥â ᮡ®î10 ¯à¨ K > 0 ¤¢¥ ¯ à ««¥«ì­ë¥ ¬­¨¬ë¥ ¯àï¬ë¥,20 ¯à¨ K < 0 ¤¢¥ ¢¥é¥á⢥­­ë¥ ¯ à ««¥«ì­ë¥ ¯àï¬ë¥,30 ¯à¨ K = 0 ¤¢¥ ᮢ¯ ¤ î騥 ¯àï¬ë¥.“⢥ত¥­¨¥ 5.1.

‚ á«ãç ¥ a1 cos α + a2 sin α = 0 ¬ë ¨¬¥¥¬µK= deta11a1a1a0¶µ+ deta22a2a2a0¶.„®ª § â¥«ìá⢮. Œë ¨¬¥¥¬µdeta11a1a1a0¶µ+ deta22a2¶a2a0= a11 a0 − a21 + a22 a0 − a22= a0 (a11 + a22 ) − a21 − a22 = a0 S − a21 − a22 .…᫨ a1 = a2 = 0, â® a02 = 0, ¨ ¨ ¢ í⮬ á«ãç ¥ ã⢥ত¥­¨¥ 5.1 ¤®ª § ­®.ãáâì a21 + a22 6= 0. Œë ¨¬¥¥¬ a01 = a1 cos α + a2 sin α = 0, ¯®í⮬ãcos α =±a2,a21 + a22sin α =p∓a1.a21 + a22p‘«¥¤®¢ â¥«ì­®,a02qa21 + a22 ⇒ (a02 )2= −a1 sin α + a2 cos α = ±= a21 + a22 .¥‘¢®©á⢮ 5.1. ‚¥«¨ç¨­ K= detµa11a1a1a0¶µ+ deta22a2a2a0¶­¥ ¬¥­ï¥âáï ¯à¨ ¯®¢®à®â¥ ª®®à¤¨­ â­ëå ®á¥©.„®ª § â¥«ìá⢮.

ãáâìµ ¶ µ¶µ ¶cosϕ − sin ϕxx~ .= sin ϕ cos ϕyy~’®£¤ , ¯à®¨§¢®¤ï ¢ F (x, y) ᮮ⢥âáâ¢ãî騥 § ¬¥­ë ¯¥à¥¬¥­­ëå, ¬ë ¯à¨å®¤¨¬ ª¢ëà ¦¥­¨îFe(~x, y~) = F (cos x~ϕ− y~ sin ϕ, sin x~ϕ +~y cos ϕ) = a~11 x~2 +2~a12 x~y~+~a22 y~2 +2~a1 x~ +2~a2 y~+~a0 ,6£¤¥ a~1 = a1 cos ϕ + a2 sin ϕ, a~2 = −a1 sin ϕ + a2 cos ϕ, a~0 = a0 ,a~11= a11 cos2 ϕ + 2a12 cos ϕ sin ϕ + a22 sin2 ϕ,a~22= a11 sin2 ϕ − 2a12 cos ϕ sin ϕ + a22 cos2 ϕ.’®£¤  ­¥á«®¦­® ¯à®¢¥à¨âì, ç⮵deta~11a~1a~1a~0¶µ+ deta~22a~2a~2a~0¶µ= deta11a1a1a0¶µ+ deta22a2a2a0¶. ¥‹¥£ª® ¢¨¤¥âì, çâ® ç¨á«® K ­¥ ¬¥­ï¥âáï ¨ ¯à¨ ®âà ¦¥­¨¨ ®¤­®© ª®®à¤¨­ â­®©®á¨ ®â­®á¨â¥«ì­® ¤à㣮©, ¯®í⮬㠮­® ï¥âáï ®¤¨­ ª®¢ë¬ ¢® ¢á¥å ¯àאַ㣮«ì­ëå á¨á⥬ å ª®®à¤¨­ â á ®¡é¨¬ ­ ç «®¬, ®¤­ ª® ¬®¦¥â ¬¥­ïâìáï ¯à¨ á¤¢¨£¥á¨áâ¥¬ë ª®®à¤¨­ â.

—¨á«® K ­ §ë¢ ¥âáï ®à⮣®­ «ì­ë¬ ᥬ¨¨­¢ à¨ ­â®¬(¯®«ã¨­¢ à¨ ­â®¬).Žà⮣®­ «ì­ë¥ ¨­¢ à¨ ­âë ªà¨¢ëå 2-£® ¯®à浪 Ž¯à¥¤¥«¥­¨¥ 5.1. Žà⮣®­ «ì­ë¬ ¨­¢ à¨ ­â®¬ ¯®«¨­®¬  P (x1 , . . . , xn ) ­ §ë¢ -¥âáï äã­ªæ¨ï ®â ª®íää¨æ¨¥­â®¢ í⮣® ¯®«¨­®¬ , ª®â®à ï ­¥ ¬¥­ï¥âáï ¯à¨ ¯¥à¥å®¤¥®â ®¤­®© ¯àאַ㣮«ì­®© á¨áâ¥¬ë ª®®à¤¨­ â ª ¤à㣮©. €ä䨭­ë¬ ¨­¢ à¨ ­â®¬¯®«¨­®¬  P (x1 , . . . , xn ) ­ §ë¢ ¥âáï äã­ªæ¨ï ®â ª®íää¨æ¨¥­â®¢ í⮣® ¯®«¨­®¬ ,ª®â®à ï ­¥ ¬¥­ï¥âáï ¯à¨ ¯¥à¥å®¤¥ ®â ®¤­®©  ä䨭­®© á¨áâ¥¬ë ª®®à¤¨­ â ª ¤à㣮©.‡ ¯¨è¥¬F (x, y ) = a11 x2 + 2a12 xy + a22 y 2 + 2a1 x + 2a2 y + a0( x y 1) a11 a12=a1a12a22a2 a1xa2   y  .a01(5.16)‘¤¥« ¥¬ ®à⮣®­ «ì­ãî § ¬¥­ã ¯¥à¥¬¥­­ëå ¢ (5.16), â.