Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 235 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 

cÈÏËã


sËãÒÓˮөËÏÈmÒ°Ò亰ÒmãÒÓˮӺä¹¯º°¯ÈÓ°mË
~ÈäËÒäº°ºmº}¹Óº°m°Ë²ºË}m
3
Λ
ã«}ºº¯©²
Ψ
(,,)
xy
10
=
Ë°¯È°
°äÈ¯ÒmÈËäÈ«ÓÈäÒãÒÓÒ«mº¯ºº¹º¯«}È¯È°¹ºãºÎËÓÓÈ«m¹¯º°¯ÈÓ°mËÓÈ¹ãº°}º°Ò
z
=
1

°m
3
Λ
m©¹ºãÓ«Ë°« ÏÈäËÓÈ ÈÏÒ°È ¹¯Ò }ºº¯º® ¹ãº°}º°
z =
1
¹Ë¯Ë²ºÒ
°ÈäÈm°Ë«sÈ®Ëäã«ªº®ÏÈäËÓ©ÈÏÒ°È¹¯ÈmÒãºÒÏäËÓËÓÒ«}ºªÁÁÒÒËÓºm}mÈ
¯ÈÒÓººÁÓ}ÒºÓÈãÈ
Ψ
(,,)
xyz

ËääÈ

lÈ¯ÒÈ
S
¹Ë¯Ë²ºÈºÈÏÒ°È
},,{
321
ggg
}ÈÏÒ°
},,{
321
ggg
′′′
¹¯Ò
}ºº¯º®¹ãº°}º°
z =
1
¹Ë¯Ë²ºÒ°ÈäÈm°Ë«ÒäËËmÒ
S =
σσσ
σσσ
11 12 13
21 22 23
001

iº}ÈÏÈËã°mº
~ÈäËÓÈ}ºº¯ÒÓÈm¹ãº°}º°Ò
Oxy
m©¹ºãÓ«Ë°«¹ºÁº¯äãÈä
xx y
yx y
=
′
+
′
+
=
′
+
′
+
σσσ
σσσ
11 12 13
21 22 23

Óº¹º°}ºã}¹¯Òªºä
z =
1
Ò
1
=
′
z
º
x
y
x
y
10011
11 12 13
21 22 23
=
′
′
σσσ
σσσ

sËm©¯ºÎËÓÓº°äÈ¯Ò©
S
°ãËËÒÏºËmÒÓºº°ãºmÒ«
det
σσ
σσ
11 12
21 22
0
≠

ËääÈº}ÈÏÈÓÈ
º°}ºã}¯ÈÓÒ°ÒÓÈ¯È}mȯÈÒÓººÁÓ}ÒºÓÈãÈÓËäËÓ«°«¹¯Òã©²
ÏÈäËÓȲÈÏÒ°È ºªºËm˯өäÒã«ÏÈäËÓ¹Ë¯Ëmº«Ò² ¹ãº°}º°
z =
1
°Èäm
°Ë«ºªºä
rg
Ψ
Ò
sgn
Ψ
°º²¯ÈÓ«°«¹¯ÒÈ}Ò²ÏÈäËÓȲÈÒ°ãÈ
rg
Ψ
Ò
sgn
Ψ
«mã«°«ÒÓmȯÒÈÓÈäÒ¯ÈmÓËÓÒ«ãÒÓÒÒmº¯ºº¹º¯«}È
È}Òäº¯ÈϺäº}ÈÏÈÓÈ
˺¯ËäÈ

¯Òã©²ÏÈäËÓȲË}ȯºmº®°Ò°Ëä©}ºº¯ÒÓÈÓÈ¹ãº°}º°Ò
Oxy
Ò°ãÈ
rg
Φ

rg
Ψ

sgn
Φ
Ò
sgn
Ψ
«mã«°«ÒÓmȯÒÈÓÈäÒãÒÓÒÒ
mº¯ºº¹º¯«}È
cÈÏËã 
sËãÒÓˮөËÏÈmÒ°Ò亰ˆÒmãÒÓˮӺ乯º°ˆ¯ÈÓ°ˆmË



         ~ÈäˈÒ䈺°ºmº}‚¹Óº°ˆ m°Ë²ˆºË}m Λ3 ã«}ºˆº¯©² Ψ ( x , y ,1) = 0 Ë°ˆ ¯È°
°äȈ¯ÒmÈËäÈ«ÓÈäÒãÒÓÒ«mˆº¯ºº¹º¯«}ȯȰ¹ºãºÎËÓÓÈ«m¹¯º°ˆ¯ÈÓ°ˆmËÓȹ㺰}º°ˆÒ
z = 1 
         
         
      ‚°ˆ  m Λ3  m©¹ºãӫˈ°« ÏÈäËÓÈ ­ÈÏÒ°È ¹¯Ò }ºˆº¯º® ¹ãº°}º°ˆ  z = 1  ¹Ë¯Ë²º҈
°ÈäÈm°Ë­«sÈ®Ëä㫪ˆº®ÏÈäËÓ©­ÈÏҰȹ¯ÈmÒãºÒÏäËÓËÓÒ«}ºªÁÁÒÒËӈºm}mÈ
¯ÈˆÒÓººÁ‚Ó}ÒºÓÈãÈΨ ( x , y , z ) 
      
      
 ËääÈ                   lȈ¯ÒÈ S ¹Ë¯Ë²ºÈºˆ­ÈÏÒ°È {g1 , g 2 , g 3 } }­ÈÏÒ°‚ {g1′ , g 2′ , g 3′ } ¹¯Ò
 
                          }ºˆº¯º®¹ãº°}º°ˆ  z = 1 ¹Ë¯Ë²º҈°ÈäÈm°Ë­«ÒäËˈmÒ
                                                                             σ 11 σ 12 σ 13
                                                                         S = σ 21 σ 22 σ 23 
                                                                              0    0    1
        
  iº}ÈÏȈËã°ˆmº
   
     ~ÈäËÓÈ}ºº¯ÒÓȈm¹ãº°}º°ˆÒ Oxy m©¹ºãӫˈ°«¹ºÁº¯ä‚ãÈä
                                                               x = σ 11 x ′ + σ 12 y ′ + σ 13
                                                                                              
                                                               y = σ 21 x ′ + σ 22 y ′ + σ 23
         
                                                        x   σ 11 σ 12                                         σ 13      x′
         Óº¹º°}ºã }‚¹¯Òªˆºä z = 1 Ò z ′ = 1 ˆº y = σ 21 σ 22                                         σ 23      y ′ 
                                                                                  1          0        0        1        1
                                                                                  
                                                                                                                                  σ 11 σ 12
         sËm©¯ºÎËÓÓº°ˆ äȈ¯Ò© S °ãË‚ˈÒϺËmÒÓºº‚°ãºmÒ« det                                                                     ≠ 0 
                                                                                                                                  σ 21 σ 22
     
     ËääȺ}ÈÏÈÓÈ
      
      
      º°}ºã }‚¯ÈÓÒ°ÒÓȈ‚¯È}mȯȈÒÓººÁ‚Ó}ÒºÓÈãÈÓËäËÓ« ˆ°«¹¯Òã ­©²
ÏÈäËÓȲ­ÈÏҰȈºªˆº­‚ˈm˯өäÒã«ÏÈäËӹ˯Ëmº«Ò²¹ãº°}º°ˆ  z = 1 °Èä‚m
°Ë­«ºªˆºä‚ rg Ψ Ò sgnΨ °º²¯ÈÓ« ˆ°«¹¯ÒˆÈ}Ò²ÏÈäËÓȲÈÒ°ãÈ rg Ψ Ò sgnΨ 
«mã« ˆ°«ÒÓmȯÒÈӈÈäÒ‚¯ÈmÓËÓÒ«ãÒÓÒÒmˆº¯ºº¹º¯«}È
       
       
       ‘È}Ò亭¯ÈϺäº}ÈÏÈÓÈ
       
 ‘˺¯ËäÈ     ¯Òã ­©²ÏÈäËÓȲË}ȯˆºmº®°Ò°ˆËä©}ºº¯ÒÓȈÓȹ㺰}º°ˆÒ Oxy 
 
                          Ò°ãÈ rg Φ  rg Ψ  sgn Φ  Ò sgnΨ  «mã« ˆ°« ÒÓmȯÒÈӈÈäÒ ãÒÓÒÒ
                          mˆº¯ºº¹º¯«}È
             
             

Страницы