L-7-Spring2018 (Грешнов - Лекции (1-15))

‹…Š–ˆŸ ü7„ â : 20.03.2018‘¢®©á⢮ 7.1. ‹î¡ ï â®çª  M½ = (x, y ), ¯à¨­ ¤«¥¦ é ï ᮢ¯ ¤ î騬 ¯àï¬ë¬= 0,a12 x + a22 y + a2 = 0.„®ª § â¥«ìá⢮. …᫨ ªà¨¢ ï 2-£® ¯®à浪  γ ï¥âáï ¯ à®© ᮢ¯ ¤ îé¨å ¯àï¬ëåAx + By + C = 0, â® ¥¥ ãà ¢­¥­¨¥ ¨¬¥¥â ¢¨¤γ,㤮¢«¥â¢®àï¥â ãà ¢­¥­¨ï¬F (x, y ) = (Ax + By + C )2a11 x + a12 y + a1= A2 x2 + 2ABy + B 2 y2 + 2ACx + 2BCy + C 2 = 0.¤«ï ­¥ª®â®àëå A, B, C ∈ R, A2 + B 2 6= 0. ’ ª¨¬ ®¡à §®¬,a11= A2 ,‘«¥¤®¢ â¥«ì­®,½a12= AB,a22= B2,a1= AC,a2= BC.0 = A(Ax + By + C ) = a11 x + a12 y + a1 ,0 = B (Ax + By + C ) = a12 x + a22 y + a2 .¥’¥®à¥¬  7.1 (® 業âॠªà¨¢®© 2-£® ¯®à浪 ). Š®«¨ç¥á⢮ 業â஢ ªà¨¢®©2-£® ¯®à浪  å à ªâ¥à¨§ã¥âáï á«¥¤ãî騬 ®¡à §®¬:10 δ 6= 0 | ®¤¨­ 業âà,20 δ = = 0 | ¯àï¬ ï «¨­¨ï 業â஢,30 δ = 0, 6= 0 | 業â஢ ­¥â.„®ª § â¥«ìá⢮.

. 10 ®ç¥¢¨¤¥­.. 20 . …᫨(a21 , a22 , a2 ) = (ka11 , ka12 , ka1 ) ¤«ï ­¥ª®â®à®£®k ∈ R,â® ¬­®¦¥á⢮ â®ç¥ª, ª®®à¤¨­ âë (x0 , y0 ) ª®â®àëå 㤮¢«¥â¢®àïîâ á¨á⥬¥ «¨­¥©­ëåãà ¢­¥­¨©½a11 x0 + a12 y0 + a1 = 0,a12 x0 + a22 y0 + a2 = 0,®¡à §ãîâ ¯àï¬ãî «¨­¨î ­  ¯«®áª®áâ¨; ªà®¬¥ ⮣®,µdeta11ka11a12ka12¶= δ = 0,a11det ka11a1a12ka12a2a1ka1  = = 0.a0Ž¡à â­®, ¯ãáâì δ = = 0.®ª ¦¥¬, çâ® δ = 0 ⇒ a211 + a222 6= 0. „¥©á⢨⥫쭮, ¥á«¨ a211 + a222 = 0 ¨ δ = 0,â® a12 = 0, çâ® ¯à®â¨¢®à¥ç¨â ⮬ã, çâ® a211 + a212 + a222 6= 0.12®í⮬㠬®¦­® ¯®« £ âì, ­¥ 㬥­ìè ï ®¡é­®áâ¨, çâ®, a11 6= 0. ’®£¤  (δ = 0)­ ©¤¥âáï ç¨á«® k ∈ R â ª®¥, çâ® a12 = ka11 , a22 = ka12 .  áªà®¥¬ ®¯à¥¤¥«¨â¥«ìa11 = det  a12a1¯® âà¥â쥩 áâப¥, ¢ १ã«ìâ â¥ ¨¬¥¥¬µ0 = = a1 detka11ka12a1a2¶µ− a2 detµa11a12ka11ka22a2a1a2a1a2 a0¶µ+ a0 kδ = (ka1 − a2 ) det¶a11 a1¨«¨ ka1 = a2 , ¨«¨ det a a= 0, çâ® á ãç¥â®¬ a12122a2 .

‘«¥¤®¢ â¥«ì­®,(a21 , a22 , a2 ) = (ka11 , ka12 , ka1 ),   §­ ç¨â,’®£¤ ka1 =­ëå ãà ¢­¥­¨©½a11a12a1a2¶.= ka11 ¢«¥ç¥âá¨á⥬  «¨­¥©-a11 x + a12 y + a1a12 x + a22 y + a2= 0,(7.1)= 0,®¯à¥¤¥«ï¥â ᮡ®© ¯àï¬ãî (業â஢) ­  ¯«®áª®áâ¨.. 30 ’ ª ª ª δ = 0, â® a211 + a222 6= 0, ¨ ¬ë ¬®¦¥¬ ¯®« £ âì, ­¥ 㬥­ìè ï®¡é­®áâ¨, çâ® a11 6= 0 (á¬. ¯.

20 ). à¥¤¯®«®¦¨¬, çâ® k(a11 , a12 ) = (a12 , a22 ) ¤«ï­¥ª®â®à®£® k 6= 0, ¨ ¯à¨ í⮬ áãé¥áâ¢ã¥â à¥è¥­¨¥ (x0 , y0 ) á¨á⥬ë (7.1). ’®£¤ ,㬭®¦ ï ¯¥à¢®¥ à ¢¥­á⢮ ¢ (7.1) ­  k ¨ ¢ëç¨â ï ®¤­® à ¢¥­á⢮ ¨§ ¤à㣮£®, ¬ë¯®«ã稬, çâ® a2 = ka1 , ®âªã¤  = 0. ‡­ ç¨â, k = 0,   íâ®, á ãç¥â®¬ áãé¥á⢮¢ ­¨ï業âà , ¢«¥ç¥â a2 = 0, ®âªã¤  = 0. ‘«¥¤®¢ â¥«ì­®, ¯à¨ ãá«®¢¨¨, çâ® δ = 0, 6= 0,㠪ਢ®© ­¥â 業â஢.¥‘¢®©á⢮ 7.1. Ž¯à¥¤¥«¥­¨¥ 業âà  ªà¨¢®© 2-£® ¯®à浪  γ ­¥ § ¢¨á¨â ®â ¢ë¡®à  ä䨭­®© á¨áâ¥¬ë ª®®à¤¨­ â.„®ª § â¥«ìá⢮. ãáâì (x0 , y0 ) | 業âà ªà¨¢®© 2-£® ¯®à浪 .

’®£¤   a11 a12 a1x00 a12 a22 a2   y0  = .0a1 a2 a01a1 x0 + a2 y0 + a0e xŒë ¨¬¥¥¬ ¤¢¥  ä䨭­ë¥ á¨áâ¥¬ë ª®®à¤¨­ â (O, x, y) ¨ (O,~, y~), á¢ï§ ­­ë¥ ᮮ⭮襭¨ï¬¨   µ ¶ µ¶µ ¶ µ ¶xc11 c12 c1xc11 c12x~ + c1 ⇔  y  =  c c c   xy~~  ,=21222yc21 c22y~c210 0 11µ£¤¥C=c11c21c12c22¶6=0. ãáâì (~x0 , y~0 ) | ª®®à¤¨­ âë 業âà  ªà¨¢®©e x䨭­®© á¨á⥬¥ ª®®à¤¨­ â (O,~, y~), â. ¥.c11 c120c12c220   c1x~0x0c2y~0 = y0  ,111γ¢  ä-3¨F (c11 x+c12 y +c1 , c21 x+c22 y +c2 ) = Fe(~x, y~) = a~11 x~2 +2~a12 x~y~+~a22 y~2 +2~a1 x~+2~a2 y~+~a0= 0.’®£¤  ¬ë ¨¬¥¥¬a~11a~12a~1 a~1x~0a~2   y~0 a~01 a11 a12 a1c11 c21 0c11 c12 c1x~0=  c12 c22 0   a12 a22 a2   c21 c22 c2   y~0 c1 c2 1a1 a2 a00 0 11 00c11 c21 0=,00=  c12 c22 0  a1 x0 + a2 y0 + a0c1 c2 1a1 x0 + a2 y0 + a0a~12a~22a~2®âªã¤  á«¥¤ã¥â ᢮©á⢮ 7.1.

®¯ãâ­® ¬ë ¤®ª § «¨ ⮦¤¥á⢮ a1 x0 + a2 y0 + a0 =a~1 x~0 + a~2 y~0 + a~0 .¥Š á â¥«ì­ë¥ ª ªà¨¢ë¬ 2-£® ¯®à浪  ©¤¥¬ â®çª¨ ¯¥à¥á¥ç¥­¨ï ¯àאַ©½l:x = x0 + αt,y = y0 + βt¨ ªà¨¢®© 2-£® ¯®à浪 γ:F (x, y ) = a11 x2 + 2a12 xy + a22 y 2 + 2a1 x + 2a2 y + a0= 0.®á«¥ ¯®¤áâ ­®¢ª¨ ãà ¢­¥­¨© ¯àאַ© ¢ ¢ëà ¦¥­¨¥ F (x, y), ¬ë ¯®«ã稬F (x0 + αt, y0 + βt) = F2 t2 + 2F1 t + F0 = 0,F2 = a11 α2 + 2a12 αβ + a22 , F1 = α(a11 x0 + a12 y0 + a1 ) + β (a12 x0 + a22 y0 + a2 ),F0= F (x0 , y0 ).’®çª¨ ¯¥à¥á¥ç¥­¨ï γ ¨ l ᮮ⢥âáâ¢ãîâ ª®à­ï¬ ª¢ ¤à â­®£® ãà ¢­¥­¨ïF2 t2 + 2F1 t + F0= 0.(7.1)…᫨ F2 6= 0, â® íâ® ãà ¢­¥­¨¥ ¨¬¥¥â ¤¢  ª®à­ï t1 , t2 |à §«¨ç­ëå ¨«¨ ᮢ¯ ¤ îé¨å(¢ í⮬ á«ãç ¥ ¢á¥£¤  ¢¥é¥á⢥­­ëå).

‚ á«ãç ¥, ª®£¤  t1 = t2 , ¯àﬠï l ­ §ë¢ ¥âá猪á â¥«ì­®© ª ªà¨¢®© γ .4„«ï ­ å®¦¤¥­¨ï ª á â¥«ì­®© 㤮¡­® §  â®çªã á ª®®à¤¨­ â ¬¨ (x0 , y0 ) ¡à âì âãâ®çªã, ª®â®à ï ¯à¨­ ¤«¥¦¨â l ∩ γ . ‚ í⮬ á«ãç ¥ F0 = F (x0 , y0 ) = 0, ¨ ¬ë ¨¬¥¥¬t(F2 t + 2F1 ) = 0, ®âªã¤  áࠧ㠦¥ ¯®«ãç ¥¬, çâ® t1 = 0. ® ¤«ï ⮣®, ç⮡ë t1 = t2 ,­¥®¡å®¤¨¬® ¢ë¯®«­¥­¨¥ ⮦¤¥á⢠t2=−2F 1F2= 0 ⇒ F1 = α(a11 x0 + a12 y0 + a1 ) + β (a12 x0 + a22 y0 + a2 ) = 0(7.2)Ž¡®§­ ç¨¬ A1 = a11 x0 + a12 y0 + a1 , A2 = a12 x0 + a22 y0 + a2 .

…᫨ ¡ë â ª á«ã稫®áì,çâ® A1 = A2 = 0, â® ¬ë ¯®¯ «¨ ¡ë ¢ á¨âã æ¨î, ª®£¤  (x0 , y0 ) | 業âà ªà¨¢®© γ ,¯à¨­ ¤«¥¦ é¨© γ (â ª ï â®çª  ­ §ë¢ ¥âáï ®á®¡®© â®çª®© ªà¨¢®© γ ). “ç¨â뢠磌 áá¨ä¨ª æ¨î ªà¨¢ëå 2-£® ¯®à浪 , ¬ë ¬®¦¥¬ ᪠§ âì, çâ® ®á®¡ë¥ â®çª¨ ¥áâì{ ¢ á«ãç ¥ δ > 0 ã ¤¢ãå ¬­¨¬ëå ¯¥à¥á¥ª îé¨åáï ¯àï¬ëå (â®çª  ¨å ¯¥à¥á¥ç¥­¨ï),{ ¢ á«ãç ¥ δ < 0 ã ¤¢ãå ¯¥à¥á¥ª îé¨åáï ¯àï¬ëå (â®çª  ¨å ¯¥à¥á¥ç¥­¨ï),{ ¢ á«ãç ¥ δ = 0 (K = 0) ã ¯ àë ᮢ¯ ¤ îé¨å ¯àï¬ëå.Š á â¥«ì­ë¥ ª ªà¨¢ë¬ 2-£® ¯®à浪  ®¯à¥¤¥«ïîâáï ¢ ­¥®á®¡ëå â®çª å,â.

¥. â ª¨å â®çª å á ª®®à¤¨­ â ¬¨ (x0 , y0 ), çâ®(a11 x0 + a12 y0 + a1 )2 + (a12 x0 + a22 y0 + a2 )2 6= 0.Œë ¨¬¥¥¬, á¬. (7.2), çâ® αA1 + βA2 = 0, á«¥¤®¢ â¥«ì­®, ­ ¯à ¢«ïî騩 ¢¥ªâ®àª á â¥«ì­®© ¯à®¯®à樮­ «¥­ ¢¥ªâ®àã (−A2 , A1 ). ’®£¤  ¬ë ¯®«ãç ¥¬ á«¥¤ãî饥ãà ¢­¥­¨¥ ª á â¥«ì­®© ª ªà¨¢®© 2-£® ¯®à浪  γ , ¯à®å®¤ï饩 ç¥à¥§ â®çªã á ª®®à¤¨­ â ¬¨ (x0 , y0 ):x − x0y − y0=⇔ (a11 x0 + a12 y0 + a1 )(x − x0 ) + (a12 x0 + a22 y0 + a2 )(y − y0 ) = 0−A2A1⇔ (a11 x0 + a12 y0 + a1 )x + (a12 x0 + a22 y0 + a2 )y + a1 x0 + a2 y0 + a0 = 0 aaax11121( x0 y0 1) ⇔a12 a22 a2y=0a1 a2 a01¯¯∂F (x, y ) ¯∂F (x, y ) ¯⇔(x − x0 ) +(y − y0 ) = 0.

(7.3)¯¯∂x∂y(x,y)=(x0 ,y0 )(x,y)=(x0 ,y0 )’ ª¨¬ ®¡à §®¬, ¢ á«ãç ¥ í««¨¯á , £¨¯¥à¡®«ë, ¯ à ¡®«ë ª á â¥«ì­ ï áãé¥áâ¢ã¥â¢ «î¡®© â®çª¥, ¯à¨­ ¤«¥¦ é¥© í⨬ ªà¨¢ë¬. ‚ á«ãç ¥ ¯¥à¥á¥ª îé¨åáï ¯àï¬ë媠á â¥«ì­ ï áãé¥áâ¢ã¥â ¢ «î¡®© â®çª¥, ¯à¨­ ¤«¥¦ é¥© ªà¨¢®©, §  ¨áª«î祭¨¥¬â®çª¨ ¯¥à¥á¥ç¥­¨ï íâ¨å ¯àï¬ëå, ¨ ᮢ¯ ¤ ¥â á ®¤­®© ¨§ íâ¨å ¯àï¬ëå. ‚ á«ãç ¥ ¯ à ««¥«ì­ëå ¯àï¬ëå ª á â¥«ì­ ï áãé¥áâ¢ã¥â ¢ «î¡®© â®çª¥, ¯à¨­ ¤«¥¦ é¥© ªà¨¢®©, ¨ ᮢ¯ ¤ ¥â á ®¤­®© ¨§ íâ¨å ¯àï¬ëå.5‚ á«ãç ¥ ᮢ¯ ¤ îé¨å ¯àï¬ëå ª á â¥«ì­ ï ­¥ áãé¥áâ¢ã¥â ­¨ ¢ ®¤­®© â®çª¥,¯à¨­ ¤«¥¦ é¥© ªà¨¢®©. „¥©á⢨⥫쭮, à áᬠâਢ ï ª ­®­¨ç¥áª®¥ ãà ¢­¥­¨¥á®¢¯ ¤ îé¨å ¯àï¬ëå x2 = 0, ¬ë ¯®«ãç ¥¬, çâ® «î¡ ï â®çª  â ª®© ªà¨¢®© |®á®¡ ï.‘¢®©á⢮ 7.2.

Ž¯à¥¤¥«¥­¨¥ ª á â¥«ì­®© ­¥ § ¢¨á¨â ®â ¢ë¡®à   ä䨭­®© á¨áâ¥¬ë ª®®à¤¨­ â.„®ª § â¥«ìá⢮. ‘¢®©á⢮ 7.2 ¢ë⥪ ¥â ¨§ ⮣®, çâ® ¯à¨ ¤¥©á⢨¨ ¯à®¨§¢®«ì­®£® ä䨭­®£® ¯à¥®¡à §®¢ ­¨ï (¢á¥£¤  ïî饣®áï ¡¨¥ªâ¨¢­ë¬ ®â®¡à ¦¥­¨¥¬) ¯àï¬ë¥ ¯¥à¥å®¤ïâ ¢ ¯àï¬ë¥, á®åà ­ï¥âáï  ä䨭­ë© ¢¨¤ ªà¨¢®© 2-£® ¯®à浪  (á¬. «¥ªæ¨î ü6), 業âà ªà¨¢®© 2-£® ¯®à浪  ¯¥à¥å®¤¨â ¢ 業âà ªà¨¢®© 2-£® ¯®à浪  (᢮©á⢮ 7.1).¥€á¨¬¯â®â¨ç¥áª¨¥ ­ ¯à ¢«¥­¨ï ª ªà¨¢ë¬ 2-£® ¯®à浪  áᬮâਬ ª¢ ¤à â­®¥ ãà ¢­¥­¨¥ (7.1).Ž¯à¥¤¥«¥­¨¥ 7.1. ¥­ã«¥¢®© ¢¥ªâ®à (α, β ) ¨¬¥¥â  á¨¬¯â®â¨ç¥áª®¥ ­ ¯à ¢«¥­¨¥ª ªà¨¢®© 2-£® ¯®à浪  γ , ¥á«¨ F2 (α, β ) = 0. €á¨¬¯â®â¨ç¥áª®¥ ­ ¯à ¢«¥­¨¥ ª ªà¨¢®© 2-£® ¯®à浪  γ | ª« áá ¢¥ªâ®à®¢, ¯à®¯®à樮­ «ì­ëå ª ª®¬ã-­¨¡ã¤ì ¢¥ªâ®àã,¨¬¥îé¥¬ã  á¨¬¯â®â¨ç¥áª®¥ ­ ¯à ¢«¥­¨¥ ª ªà¨¢®© 2-£® ¯®à浪  γ .‘¢®©á⢮ 7.3.

Ž¯à¥¤¥«¥­¨¥  á¨¬¯â®â¨ç¥áª®£® ­ ¯à ¢«¥­¨ï ­¥ § ¢¨á¨â ®â ¢ë-¡®à   ä䨭­®© á¨áâ¥¬ë ª®®à¤¨­ â, ¢ ª®â®à®© § ¤ ­® ãà ¢­¥­¨¥ ªà¨¢®© γ .„®ª § â¥«ìá⢮. Œë ¨¬¥¥¬F2 (α, β ) = ( αβ)µa11a12a12a22¶µ ¶α.βe x áᬮâਬ ªà¨¢ãî γ ¢ ¤à㣮©  ä䨭­®© á¨á⥬¥ ª®®à¤¨­ â (O,~, y~), â. ¥. § ¬¥­¨¬¯¥à¥¬¥­­ë¥ (x, y) ¯® ¯à ¢¨«ãµ ¶ µx= cc11y21µ£¤¥C=c11c21c12c22  ¶µ ¶ µ ¶c12x~ + c1 ⇔  xy  =  cc1121c22y~c201 c1x~c2y~  ,c12c22011¶6=0, ¨ à áᬮâਬ ãà ¢­¥­¨¥ Fe(~x, y~) = 0 ªà¨¢®© γ . ’®£¤ , ­ «®£¨ç­® (7.1), ®¯à¥¤¥«ï¥¬ ª¢ ¤à â­®¥ ãà ¢­¥­¨¥ Fe2 t2 + 2Fe1 t + Fe0 = 0, £¤¥µ~ β~ ) a~11 a~12Fe2 (~α, β~) = ( αa~12 a~22Œë ¨¬¥¥¬µa~11a~12a~12a~22¶µ=c11c12¶µ ¶α~ ,β~c21c22¶µa11a12µ ¶ µαc11=βc21a12a22¶µc11c21c12c22c12c22¶µ ¶α~ .β~¶,6®âªã¤  ¨ á«¥¤ã¥â ᢮©á⢮ 7.3.¥‘¢®©á⢮ 7.4. …᫨ ¯àﬠï l ¨¬¥¥â  á¨¬¯â®â¨ç¥áª®¥ ­ ¯à ¢«¥­¨¥, â® ®­  ¨«¨æ¥«¨ª®¬ «¥¦¨â ¢ ªà¨¢®© γ , ¨«¨ ¨¬¥¥â á ­¥© ­¥ ¡®«¥¥ ®¤­®© ®¡é¥© â®çª¨.„®ª § â¥«ìá⢮.

ãáâì (α, β ) | ­ ¯à ¢«ïî騩 ¢¥ªâ®à ¯àאַ© l, ⮣¤  F2 = 0, ¨¬ë ¨¬¥¥¬ 2F1 t + F0 = 0. ®­ïâ­®, çâ® ¥á«¨ F1 6= 0, â® ¬­®¦¥á⢮ l ∩ γ á®á⮨⠨§¥¤¨­á⢥­­®© â®çª¨; ¥á«¨ F1 = 0, â® ¨«¨ l ⊂ γ (F0 = 0), ¨«¨ l ∩ γ = ∅ (F0 6= 0). ¥ˆ§ã稬 ãá«®¢¨¥ F2 (α, β ) = a11 α2 + 2a12 αβ + a22 β 2 = 0. à¥¤¯®«®¦¨¬ a11 =6 0.’®£¤  β 6= 0 (¨­ ç¥ ¯®«ã稬, çâ® α = 0).  áᬮâਬ ª¢ ¤à â­®¥ ãà ¢­¥­¨¥a11Œë ¨¬¥¥¬αβ=¡ α ¢2α+ 2a12ββ−a12 ±+ a22 = 0.pa212 − a11 a22a11=√−a12 ± −δ.a11(7.4)√−a12 ± −δ.a22(7.5)‚ á«ãç ¥ a22 6= 0  ­ «®£¨ç­® (7.4) ®¯à¥¤¥«ï¥âáï ®â­®è¥­¨¥βα=−a12 ±pa212 − a11 a22a22=…᫨ ¦¥ a11 = a22 = 0, â®, ãç¨â뢠ï ãá«®¢¨¥ a211 + a212 + a222 6= 0, ¬ë ¨¬¥¥¬a2122a12 αβ = 0,= −δ 6= 0.(7.6)ˆ§ (7.6) ¢ë⥪ ¥â, çâ® ¬ë ¨¬¥¥¬ ¤¢   á¨¬¯â®â¨ç¥áª¨å ­ ¯à ¢«¥­¨ï: (0, 1), (1, 0).’¥®à¥¬  7.2 (®¡  á¨¬¯â®â¨ç¥áª¨å ­ ¯à ¢«¥­¨ïå ªà¨¢®© 2-£® ¯®à浪 ).Šà¨¢ ï 2-£® ¯®à浪  γ ¨¬¥¥â ¨¬¥¥â ¤¢  à §«¨ç­ëå ¢¥é¥á⢥­­ëå ¨«¨ ¬­¨¬ëå,¨«¨ ¤¢  ᮢ¯ ¤ îé¨å  á¨¬¯â®â¨ç¥áª¨å ­ ¯à ¢«¥­¨ï:{ δ > 0 | ¤¢  à §«¨ç­ëå ¬­¨¬ëå,{ δ < 0 | ¤¢  à §«¨ç­ëå ¢¥é¥á⢥­­ëå,{ δ = 0 | ¤¢  ᮢ¯ ¤ îé¨å ¢¥é¥á⢥­­ëå.„®ª § â¥«ìá⢮.

 ¬ ®áâ «®áì à §®¡à âì á«ãç ©, ª®£¤  δ = 0. …᫨ a11 a22 6= 0(¢ í⮬ á«ãç ¥ δ = 0 ⇒ a12 6= 0), â® ¨á¯®«ì§ãï ä®à¬ã«ë (7.4), (7.5), ¬ë ¯®«ãç ¥¬,çâ® ªà¨¢ ï 2-£® ¯®à浪  γ ¨¬¥¥â ¤¢  ᮢ¯ ¤ îé¨å  á¨¬¯â®â¨ç¥áª¨å ­ ¯à ¢«¥­¨ï(α, β ) :αaa= − 12 = − 22 ;βa11a12¥á«¨ ¦¥, ­ ¯à¨¬¥à, a22 = 0 (⇒ a12 = 0), â® ¯® ä®à¬ã«¥ (7.4) ¬ë ¯®«ãç ¥¬αβ=−a12a11= 0 ⇒ α = 0,7  ¨§ ä®à¬ã«ë (7.5) ¬ë ¯®«ãç ¥¬ ­¥®¯à¥¤¥«¥­­®áâ좠¥âáï á«ãç © a11 = 0.αβ= 00 . €­ «®£¨ç­® à áᬠâਥŽ¯à¥¤¥«¥­¨¥ 7.2.

€á¨¬¯â®â®© ªà¨¢®© 2-£® ¯®à浪  γ ­ §ë¢ ¥âáï ¯àï¬ ï  á¨¬¯â®â¨ç¥áª®£® ­ ¯à ¢«¥­¨ï, ¯à®å®¤ïé ï ç¥à¥§ 業âà ªà¨¢®© γ .’¥®à¥¬  7.3 (®¡  á¨¬¯â®â å ªà¨¢ëå 2-£® ¯®à浪 ). Šà¨¢ë¥ 2-£® ¯®à浪 å à ªâ¥à¨§ãîâáï ç¨á«®¬ ᢮¨å  á¨¬¯â®â á«¥¤ãî騬 ®¡à §®¬:10 £¨¯¥à¡®«  ¨ ¯ à  ¯¥à¥á¥ª îé¨åáï ¯àï¬ëå ¨¬¥îâ ¤¢¥  á¨¬¯â®âë,20 ¯ à ««¥«ì­ë¥ ¯àï¬ë¥ (à §«¨ç­ë¥, ᮢ¯ ¤ î騥 ¨«¨ ¬­¨¬ë¥) ¨¬¥îâ ®¤­ã á¨¬¯â®âã,30 ®áâ «ì­ë¥ «¨­¨¨ ¢â®à®£® ¯®à浪   á¨¬¯â®â ­¥ ¨¬¥îâ.„®ª § â¥«ìá⢮. ’¥®à¥¬  7.3 ¢ë⥪ ¥â ¨§ ⥮६ 7.1, 7.2.

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