Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 234 стр.

UptoLike

Составители: 

Рубрика: 


Ë} ÒÒ}ÈÁË ¯©m©° Ë®äÈËäÈÒ}Òlnj
ÙkÓÈãÒÒ˰}È«˺äË¯Ò«ÒãÒÓË®ÓÈ«ÈãË¯Èµ

äÓºmkp
jÓmȯÒÈÓ©ãÒÓÒ®mº¯ºº¹º¯«}ÈÓÈ¹ãº°}º°Ò
sËÏÈmÒ°Ò亰ÏÓÈËÓÒ®¯ÈÓÈÒ°ÒÓÈ¯©}mȯÈÒÓººÁÓ}ÒºÓÈãÈºm©
º¯ÈÈÏÒ°È¹ºÏmºã«Ëm©¹ºãÓÒ}ãȰ°ÒÁÒ}ÈÒãÒÓÒ®mº¯ºº¹º¯«}ÈÓÈ¹ãº°}º°Ò
°¹º°ººäºãÒÓ©äº¹¯ÒmËËÓÓººm˺¯ËäË
cȰ°äº¯Òä ãÒÓÒ mº¯ºº ¹º¯«}È ÓÈ ¹ãº°}º°Ò
Oxy
m ÈÏÒ°Ë
},{
21
gg
Ò°
ÓÈÈãºä}ºº¯ÒÓÈmº}Ë
O
wÈãÒÓÒ«mºËä°ãÈËÏÈÈË°«°ºãȰӺº¹¯ËËãËÓÒ
¯ÈmÓËÓÒËämÒÈ
0222
22
=+++++ FEyDxCyBxyAx

ËÒ°ãÈ
A

B

C

D

F
Ò
E
¹¯ºÒÏmºãÓ©°ºÓÒäãÒº¯ÈÓÒËÓÒËäº
A

B
Ò
C

ÓË¯ÈmÓ©ÓãºÓºm¯ËäËÓÓº
ABC++>0

sË¯Óº¹¯ºm˯Òº¹¯ÒÏÈäËÓËÓÈÈãÈ}ºº¯ÒÓÈ}ºªÁÁÒÒËÓ©
A

B
Ò
C
ÓË äËÓ«°« È ¹¯Ò °äËÓË ÈÏÒ°È ¹¯Ëº¯ÈÏ°« }È} }ºªÁÁÒÒËÓ© }mȯÈÒÓºº
ÁÓ}ÒºÓÈãÈ °ä ˺¯Ëä  ºªºä äºÎÓº °ÒÈ º äÓººãËÓ
Ax Bxy Cy
22
2++
ÏÈÈË}mȯÈÒÓ©®ÁÓ}ÒºÓÈã
Φ
(,)x y Ax Bxy Cy=+ +
22
2
°äÈ
¯ÒË®
AB
BC
mÒ°²ºÓºäÈÏÒ°Ë
{, }gg
12

j°¹ºãÏ«m˯ÎËÓÒ«˺¯ËäÒ
rg
Φ
¯ÈÓÒ
sgn
Φ
°ÒÓÈ¯È
}mȯÈÒÓººÁÓ}ÒºÓÈãÈ
Φ
(,)
xy
ÓËÏÈmÒ°«ºm©º¯È°Ò°Ëä©}ºº¯ÒÓÈÒ°ãË
ºmÈËãÓº
rg
Φ
Ò
sgn
Φ
«mã«°« ÒÓmȯÒÈÓÈäÒ ãÒÓÒÒ mº¯ºº ¹º¯«}È ÓÈ
¹ãº°}º°Ò j°¹ºãϺmÈÓÒË äºã« °ÒÓÈ¯© Ó˺²ºÒäº ¹º°}ºã} ºÓºm¯ËäËÓÓºË
ÒÏäËÓËÓÒË ÏÓÈ}ºm m°Ë² }ºªÁÁÒÒËÓºm ¯ÈmÓËÓÒ« ãÒÓÒÒ mº¯ºº ¹º¯«}È ÒÏäËÓÒ
˰˰mËÓÓº°Èäº¯ÈmÓËÓÒË²º«ãÒÓÒ«¹¯Òªºäº°ÈÓË°«º®ÎË
º°}ºã}mÏȹҰ¯ÈmÓËÓÒ«ãÒÓÒÒmº¯ºº¹º¯«}ÈÓÈ¹ãº°}º°Òm²º«È}ÎË
Ò}ºªÁÁÒÒËÓ©
D

F
Ò
E
º°ãËËm©«°ÓÒÓË°˰mãÒº¹ºãÓÒËãÓ©ËÒÓ
mȯÒÈÓ©º¯ÈϺmÈÓÓ©ËÒÏm°Ë®°ºmº}¹Óº°Ò}ºªÁÁÒÒËÓºm
A

B

C

D

F
Ò
E

iã«ªºº¯È°°äº¯Òäm°¹ºäºÈËãÓ©®}mȯÈÒÓ©®ÁÓ}ÒºÓÈãm
3
Λ
mÒÈ
Ψ
(,,)
x y z Ax Bxy Cy Dxz Eyz Fz
=+ ++ + +
22 2
222
°äÈ¯ÒË®
ABD
BCE
DEF
mÈÏÒ°Ë
{, , }
ggg
123

 Ë }  Ò Ò    } È Á Ë  ¯ ©   m © °  Ë ®   ä È ˆ Ë ä È ˆ Ò } Ò   l n ‘ j 
ÙkÓÈã҈Ò˰}È«˺äˈ¯Ò«ÒãÒÓË®ÓÈ«ÈãË­¯ÈµäÓºmkp



jÓmȯÒÈӈ©ãÒÓÒ®mˆº¯ºº¹º¯«}ÈÓȹ㺰}º°ˆÒ
         
         
         
         sËÏÈmÒ°Ò亰ˆ ÏÓÈËÓÒ®¯ÈÓÈÒ°ÒÓȈ‚¯©}mȯȈÒÓººÁ‚Ó}ÒºÓÈãȺˆm©
­º¯È­ÈÏҰȹºÏmºã«Ëˆm©¹ºãÓ҈ }ãȰ°ÒÁÒ}ÈÒ ãÒÓÒ®mˆº¯ºº¹º¯«}ÈÓȹ㺰}º°ˆÒ
°¹º°º­ºäºˆãÒө予¹¯ÒmËËÓÓººmˆËº¯ËäË
         
         cȰ°äºˆ¯Òä ãÒÓÒ  mˆº¯ºº ¹º¯«}È ÓÈ ¹ãº°}º°ˆÒ Oxy  m ­ÈÏÒ°Ë {g1 , g 2 }  Ò °
ÓÈÈãºä}ºº¯ÒÓȈmˆº}ËOwˆÈãÒÓÒ«mº­Ëä°ã‚ÈËÏÈÈˈ°«°ºãȰӺº¹¯ËËãËÓÒ 
‚¯ÈmÓËÓÒËämÒÈ
        
                                           Ax 2 + 2 Bxy + Cy 2 + 2 Dx + 2 Ey + F = 0 
      
ËÒ°ãÈ A  B C D FÒ E¹¯ºÒÏmºã Ó©°ºÓÒäãÒ                                   º¯ÈÓÒËÓÒË䈺 A BÒ C
Ó˯ÈmÓ©ӂã ºÓºm¯ËäËÓÓº A + B + C > 0 
      
      
      sˈ¯‚Óº¹¯ºm˯҈ ˆº¹¯ÒÏÈäËÓËÓÈÈãÈ}ºº¯ÒÓȈ}ºªÁÁÒÒËӈ© A  BÒ C
ÓË äËÓ« ˆ°« È ¹¯Ò °äËÓË ­ÈÏÒ°È ¹¯Ëº­¯Èς ˆ°« }È} }ºªÁÁÒÒËӈ© }mȯȈÒÓºº
Á‚Ó}ÒºÓÈãÈ °ä ˆËº¯Ëä‚   ºªˆºä‚ äºÎÓº °҈Ȉ  ˆº äÓººãËÓ
Ax 2 + 2 Bxy + Cy 2 ÏÈÈˈ}mȯȈÒÓ©®Á‚Ó}ÒºÓÈã Φ ( x , y ) = Ax 2 + 2 Bxy + Cy 2 °äȈ
         A B
¯ÒË®          mÒ°²ºÓºä­ÈÏÒ°Ë {g1 , g 2 } 
         B C
      
      j°¹ºã ς«‚ˆm˯ÎËÓÒ«ˆËº¯ËäÒ rg Φ ¯ÈÓÒ sgn Φ °ÒÓȈ‚¯È
}mȯȈÒÓººÁ‚Ó}ÒºÓÈãÈ Φ ( x , y ) ÓËÏÈmÒ°«ˆºˆm©­º¯È°Ò°ˆËä©}ºº¯ÒÓȈÒ°ãË
ºmȈËã Óº rg Φ  Ò sgn Φ  «mã« ˆ°« ÒÓmȯÒÈӈÈäÒ ãÒÓÒÒ mˆº¯ºº ¹º¯«}È ÓÈ
¹ãº°}º°ˆÒ j°¹ºã ϺmÈÓÒË äº‚ã« °ÒÓȈ‚¯© Ó˺­²ºÒäº ¹º°}ºã }‚ ºÓºm¯ËäËÓÓºË
ÒÏäËÓËÓÒË ÏÓÈ}ºm m°Ë² }ºªÁÁÒÒËӈºm ‚¯ÈmÓËÓÒ« ãÒÓÒÒ mˆº¯ºº ¹º¯«}È ÒÏäËÓ҈
˰ˆË°ˆmËÓÓº°È亂¯ÈmÓËÓÒ˲ºˆ«ãÒÓÒ«¹¯Òªˆºäº°ˆÈÓˈ°«ˆº®ÎË
      
      
      º°}ºã }‚mÏȹҰ ‚¯ÈmÓËÓÒ«ãÒÓÒÒmˆº¯ºº¹º¯«}ÈÓȹ㺰}º°ˆÒm²º«ˆˆÈ}ÎË
Ò}ºªÁÁÒÒËӈ©DFÒEˆº°ãË‚ˈm©«°Ó҈ Ó˰‚Ë°ˆm‚ ˆãÒº¹ºãÓ҈Ëã Ó©ËÒÓ
mȯÒÈӈ©º­¯ÈϺmÈÓÓ©ËÒÏm°Ë®°ºmº}‚¹Óº°ˆÒ}ºªÁÁÒÒËӈºmABCDFÒE
      
          i㫪ˆºº¯È°°äºˆ¯Òäm°¹ºäºȈËã Ó©®}mȯȈÒÓ©®Á‚Ó}ÒºÓÈãm Λ3 mÒÈ
          
                                Ψ ( x , y , z ) = Ax 2 + 2 Bxy + Cy 2 + 2 Dxz + 2 Eyz + Fz 2 
          
                      A    B D
°äȈ¯ÒË® B          C         E m­ÈÏÒ°Ë {g1 , g 2 , g 3 } 
                     D E         F