L-13-Spring2018 (826550), страница 2
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¡à â®, ¢á类¥ ¯à ¢«¥¨¥ (α, β, γ ), 㤮¢«¥â¢®àïî饥 (13.10), ¥áâì £« ¢®¥ ¯à ¢«¥¨¥, ¯à¨ç¥¬®á®¡®¥, ¥á«¨ λ = 0 ¨ ⮫쪮 ¢ í⮬ á«ãç ¥. ¨á⥬ (13.10) ¬®¦¥â ¡ëâì § ¯¨á ª ª a11 − λa12a13α0 a21a22 − λa23β = 0.a31a32a33 − λγ0 ª¨¬ ®¡à §®¬, £« ¢ë¥ ¯à ¢«¥¨ï ¯®¢¥àå®á⨠2-£® ¯®à浪 | ¢ â®ç®á⨡ §¨á ª ®¨ç¥áª®© á¨áâ¥¬ë ª®®à¤¨ â (⮩ ¯àאַ㣮«ì®© á¨áâ¥¬ë ª®®à¤¨ â, ¢ª®â®à®© ¯®¢¥àå®áâì ¨¬¥¥â ª ®¨ç¥áª¨© ¢¨¤).
ਠí⮬ á«ãç © λ = 0 ᮮ⢥âáâ¢ã¥â ®á®¡®¬ã ¯à ¢«¥¨î, á«ãç © λ 6= 0 | ¥®á®¡®¬ã ¯à ¢«¥¨î.à⮣® «ìë¥ ¨¢ ਠâë ãà ¢¥¨ï ¯®¢¥àå®á⨠2-£® ¯®à浪 áᬮâਬ ãà ¢¥¨¥F (x, y, z )= a11 x2 + a22 y2 + a33 z 2 + 2a12 xy + 2a23 yz + 2a13 zx + 2a1 x + 2a2 y + 2a3 z + a0= 0,(13.1)¯®¢¥àå®á⨠2-£® ¯®à浪 , £¤¥ a11 2 + a222 + a233 + a212 + a223 + a213 6= 0,A1a11= a12a13a12a22a23a13a23 ,a33a11 a12A=a13a1a12a22a23a2a13a23a33a3a1a2 ,a3a0 , = det A,δ= det A1 .ãáâì | æ¥âà «ì ï ¯®¢¥àå®áâì, (x0 , y0 , z0 ) | ª®®à¤¨ âë ¥¥ æ¥âà .
¥à¥©¤¥¬ ª ®¢ë¬ ª®®à¤¨ â ¬ x~, y~, z~ ¯® ¯à ¢¨«ã~ + x0 ,x=xy = y~ + y0 ,z = z~ + z0 ,⮣¤ , ¨á¯®«ì§ãï ãà ¢¥¨ï, ®¯à¥¤¥«ïî騥 æ¥âà (x0 , y0 , z0 ), ¬ë ¯®«ãç ¥¬F (~x + x0 , y~ + y0 , z~ + z0 ) = ϕ(~x, y~, z~) + F (x0 , y0 , z0 ) = 0.(13.12)e , δ~ | § 票ï ᮮ⢥âáâ¢ãîé¨å ®¯à¥¤¥«¨â¥«¥© ¤«ï ãà ¢¥¨ï (13.12).ãáâì ᯮ«ì§ãï ¯. 20 ⥮६ë 10.2 ¨§ «¥ªæ¨¨ ü10, ¬ë ¯®«ãç ¥¬a11a21e = det a310a12a22a320a13a23a330000F (x0 , y0 , z0 ) = = F (x0 , y0 , z0 )δ.7 ª¨¬ ®¡à §®¬, ¬¨ ãáâ ®¢«¥ á«¥¤ãîé 葉६ 13.1. ¯àאַ㣮«ì®© ýᤢ¨ã⮩þ á¨á⥬¥ ª®®à¤¨ â á æ¥â஬ ¢æ¥âॠᨬ¬¥âਨ æ¥âà «ì®© ¯®¢¥àå®á⨠2-£® ¯®à浪 ¥¥ ãà ¢¥¨¥ ¨¬¥¥â¢¨¤ ϕ(~x, y~, z~) + δ = 0.¢¥¤¥¬ ®¡®§ 票ïI1= a11 + a22 + a33 ,I2= detI3= δ,a11a12µa12a22I4 = .¶µ+ deta11a13a13a33¶µ+ deta22a23a23a33¶,¥®à¥¬ 13.2.
¥«¨ç¨ë I1 {I4 ïîâáï ®à⮣® «ì묨 ¨¢ ਠ⠬¨ ãà ¢-¥¨ï ¯®¢¥àå®á⨠2-£® ¯®à浪 .®ª § ⥫ìá⢮. ®, çâ® ¢¥«¨ç¨ë I3 , I4 ïîâáï ®à⮣® «ì묨 ¨¢ ਠ⠬¨ãà ¢¥¨ï (13.1), 㦥 ¡ë«® ¤®ª § ® ¢ ⥮६¥ 10.2 «¥ªæ¨¨ ü10. ®ª ¦¥¬, ç⮢¥«¨ç¨ë I1 , I2 ïîâáï ®à⮣® «ì묨 ¨¢ ਠ⠬¨ ãà ¢¥¨ï (13.1). áᬮâਬ ãà ¢¥¨¥A1 u = λu, u ∈ R3 .(13.3)ãáâìC â u.C=c11 c21c31c12c22c32c13c23 ∈ O(3).c33¢¥¤¥¬ ®¢ë¥ ¯¥à¥¬¥ë¥u= Cu⇔ u=¥à¥¯¨è¥¬ ãà ¢¥¨¥ (13.3) ¢ ®¢ëå ¯¥à¥¬¥ëåC â A1 u = C â λu ⇐⇒ C â A1 u = λC â u ⇐⇒ C â A1 CC â u = λC â u ⇐⇒ A1 u= λu ,®âªã¤ ¢ë⥪ ¥â, çâ® ã ᨬ¬¥âà¨ç¥áª¨å ¬ âà¨æ A1 ¨ C â A1 C ®¤¨ ¨ ⥠¦¥ ᮡáâ¢¥ë¥ ç¨á« .
¤¥« ¥¬ ää¨ãî ®à⮣® «ìãî § ¬¥ã ¯¥à¥¬¥ëå xc11 y = c21c31zc12c22c32 c1c13xc23y+ c2 c33zc3¢ ãà ¢¥¨¨ (13.1), ¢ १ã«ìâ ⥠¯®«ã稬F (c11 x + c12 y + c13 z + c1 , c21 x + c22 y + c23 z + c2 , c31 x + c32 y + c33 z + c3 )= F (x , y , z ) cccaaacccx111213111213111213(x y z ) =c21 c22 c23a12 a22 a23c21 c22 c23y c31 c32 c33a13 a23 a33c31 c32 c33z cccx111213(2a1 2a2 2a3 ) y + a0 .+c21 c22 c23zc31 c32 c338 c13a11 a12 a13c11 c12 c13c23 a12 a22 a23 c21 c22 c23 = A1 . ª ¡ë«®c33a13 a23 a33c31 c32 c33¤®ª § ® ¢ëè¥, ã ¬ âà¨æ A1 ¨ A1 ®¤¨ ¨ ⥠¦¥ å à ªâ¥à¨áâ¨ç¥áª¨¥ ª®à¨ λ1 , λ2 , λ3 ,c11¡®§ 稬 c21c31c12c22c32¯®í⮬ãλ3 − I 1 λ2 + I 2 λ − δ= − det(A − λE ) = (λ − λ1 )(λ − λ2 )(λ − λ3 )= − det(A − λE ) = λ3 − I1 λ2 + I2 λ − δ . (13.4)(¥à¢®¥ ⮦¤¥á⢮ ¨§ (13.4) ¯à®¢¥àïîâáï ¥¯®á।á⢥®.) ®£¤ λ3 − I1 λ2 + I2 λ − I3= λ3 − I1 λ2 + I2 λ − I3 ⇒ Ii = Ii ,i = 1, 2, 3.¥à⮣® «ìë¥ ¯®«ã¨¢ ਠâë ãà ¢¥¨ï ¯®¢¥àå®á⨠2-£® ¯®à浪 ¡®§ 稬µK1K2K3¶µ¶µ¶a11 a1a22 a2a33 a13= det a a + det a a + det a,102031 a0 a11 a12 a1a22 a23 a2a11 a13= a12 a22 a2 + a23 a33 a3 + a13 a33a1 a2 a0a2 a3 a0a1 a3= = I4 .a1a3 ,a0 ¬¥ã ¯¥à¥¬¥ë寥६¥ë¥(x , y , z ) ¡ã¤¥¬ §ë¢ âì ®¤®à®¤ (x, y, z ) ®©, ¥á«¨xc11 y = c21zc31c12c22c32¥®à¥¬ 13.3.
¥«¨ç¨ëc13xc23y ,c33zc11det c21c31c12c22c32c13c23 6= 0.c33K1 , K2¨¢ ਠâë ¯à¨ ®¤®à®¤ëå ®à⮣® «ìëå§ ¬¥ å ¯¥à¥¬¥ëå ¢¥ªâ®à®£® ¯à®áâà á⢠R3 .®ª § ⥫ìá⢮. ®ª ¦¥¬ ⥮६ã 13.3 ¤«ï K1 . ë ¨¬¥¥¬K1= a0 I1 − (a21 + a22 + a23 ).ਠ®¤®à®¤ëå ®à⮣® «ìëå § ¬¥ å ¯¥à¥¬¥ëå ¢¥ªâ®à®£® ¯à®áâà á⢠Rn ,â. ¥. xc11 y = c21zc31c12c22c32 c13xy ,c23zc33c11 c21c31c12c22c32c13c23 = C ∈ O(3),c339¢¥«¨ç¨ a0 I1 ¥ ¬¥ï¥âáï (íâ® á«¥¤ã¥â ¨§ ⥮६ë 13.2). ë ¨¬¥¥¬c11a = c21c31c12c22c32 c13a1a1c23 a2 = a2 = a .c33a3a3 ª ª ª C ∈ O(3), â®a21 + a22 + a23= ha, ai = hC a, C ai = ha , a i = (a1 )2 + (a2 )2 + (a3 )2 .¥¬ á ¬ë¬ â¥®à¥¬ 13.3 ¤«ï K1 ¤®ª § .
«ï K2 ⥮६ 13.3 ¤®ª §ë¢ ¥âáï â®ç®â ª¦¥.¥ ¥¥ ¡ë«® ãáâ ®¢«¥®, çâ® ¯®¢¥àå®á⨠2-£® ¯®à浪 ª ®¨ç¥áª®£® ¢¨¤ ¬®¦® à §¡¨âì á«¥¤ãî騥 ¯ïâì ⨯®¢:λ1 x2 + λ2 y 2 + λ3 z 2 + τλ1 x2 + λ2 y 2 + 2νz = 0,λ1 x2 + λ2 y 2 + τ = 0,λ1 x2 + 2νy = 0,λ1 x2 + τ = 0.¥®à¥¬ (®¤¥®¢ ). ¥«¨ç¨ = 0,(I)(II)(III)(IV)(V)ï¥âáï ®à⮣® «ì묨 ¨¢ ਠ⮬â¥å ¯®¢¥àå®á⥩ 2-£® ¯®à浪 , ª®â®àë¥ ¢ ¯®¤å®¤ïé¨å ¯àאַ㣮«ìëå á¨á⥬ 媮®à¤¨ â ¨¬¥îâ ⨯ (III){(V); ¢¥«¨ç¨ K1 ï¥âáï ®à⮣® «ì묨 ¨¢ ਠ⮬ â¥å ¯®¢¥àå®á⥩ 2-£® ¯®à浪 , ª®â®àë¥ ¢ ¯®¤å®¤ïé¨å ¯àאַ㣮«ìëåá¨á⥬ å ª®®à¤¨ â ¨¬¥îâ ⨯ (V).®ª § ⥫ìá⢮. ® ⥮६¥ 13.3 ¢¥«¨ç¨ K2 ¨¢ ਠ⠯ਠ®¤®à®¤ëå ®à⮣® «ìëå § ¬¥ å ¯¥à¥¬¥ëå.
áᬠâਢ ï ãà ¢¥¨ï (III){(V) ª ª ãà ¢¥¨ï®â ¤¢ãå ¯¥à¥¬¥ëå, ¬ë ¢¨¤¨¬, çâ® ¤«ï ¨å ¢¥«¨ç¨ K2 ᮢ¯ ¤ ¥â á ᮮ⢥âáâ¢ãî騬 ®à⮣® «ìë¬ ¨¢ ਠ⮬ , ª®â®àë© ¥ ¥áâ¥á⢥® ¬¥ï¥âáï ¯à¨á¤¢¨£ å. ®í⮬㠢¥«¨ç¨ K2 ï¥âáï ®à⮣® «ì묨 ¨¢ ਠ⮬ â¥å ¯®¢¥àå®á⥩ 2-£® ¯®à浪 , ª®â®àë¥ ¢ ¯®¤å®¤ïé¨å ¯àאַ㣮«ìëå á¨á⥬ å ª®®à¤¨ ⨬¥îâ ¢¨¤ (III){(V).
«ï ¢¥«¨ç¨ë K1 à áá㦤¥¨¥ «®£¨ç®¥.¥K2§ ⥮६ 13.2, 13.3 ¨ â¥®à¥¬ë ®¤¥®¢ ¢ë⥪ îâ á«¥¤ãî騥 ¯à¨§ ª¨ ¤«ï¯®¢¥àå®á⥩ à §«¨çëå ⨯®¢:¯à¨§ ª ¯®¢¥àå®á⨠⨯ (I): I3 = δ = λ1 λ2 λ3 6= 0,¯à¨§ ª ¯®¢¥àå®á⨠⨯ (II): I3 = 0, K3 = −λ1 λ2 ν 2 6= 0,10¯à¨§ ª ¯®¢¥àå®á⨠⨯ (III): I3 = K3 = 0, I2 = λ1 λ2 6= 0,¯à¨§ ª ¯®¢¥àå®á⨠⨯ (IV): I3 = K3 = I2 = 0, K2 = −λ1 ν 2 6= 0,¯à¨§ ª ¯®¢¥àå®á⨠⨯ (IV): I3 = K3 = I2 = K2 = 0.«¥¤á⢨¥ 13.1. ®¢¥àå®áâ¨ à §«¨çëå ⨯®¢ ¬¥¦¤ã ᮡ®© ®à⮣® «ì® ¥íª¢¨¢ «¥âë..