Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 89 стр.

UptoLike

ВУЗ: 

МФТИ | Москва

Составители: 

Рубрика: 

cÈÏËã


sËãÒÓˮөËºË}©ÓÈ¹ãº°}º°ÒÒm¹¯º°¯ÈÓ°mË
jÏ°ãºmÒ«
′
=B 0
°ãËËº
22 20BACcos ( ) sin
αα
−− =
Òº}ºÓÈËãÓº
tg ; arctg2
21
2
2
αα
=
−
=
−
B
AC
B
AC
¹¯Ò
AC>

ÒãÒÎË
α
π
=
4
¹¯Ò
AC=
ºË°Ò°}ºä©®ºãÓÈ®ËÓ
°
º}ÈÎËäº¹¯ÒÈ}º®ÏÈäËÓË}ºº¯ÒÓÈmËãÒÒÓÈ
∆
ÓËÒÏäËÓÒ°«iË®°mÒ
ËãÓº ÒÏ °ººÓºËÓÒ®
121
2
1
2
22
2
+=+
−
=ctg
sin
()
α
α
AC
B
Ò Ó˯ÈmËÓ°mÈ
0
4
<≤
α
π
¹ºãÈËä
sin
()
;cos
()
2
2
4
2
4
22 22
αα
=
+−
=
−
+−
B
BAC
AC
BAC

sÈ®Ëä˹˯ÏÓÈËÓÒ«
′
A
Ò
′
C
jÏ¹¯Ë©Ò²°ººÓºËÓÒ®ÒäËËä
′
=
+
++
−
=
+
+
−
+=
=
+
+
−−
+−
+
+−
=
=
+
++−
AA B C
AC AC
B
AC AC AC
BAC
B
B
BAC
AC
BAC
12
2
2
12
222
22
22
4
2
4
2
1
2
4
2222
22
cos
sin
cos
cos sin
() ()
().
α
α
α
αα
kÓÈãºÒÓº¹ºãÈËäº
′
=
+
−+−C
AC
BAC
2
1
2
4
22
()

|}È°ãËËº
′
=
′
′
=
′′
=
+
−+−=
=−= =
∆
∆
det ( ( ) )
det ,
()
A
C
AC
AC
BAC
AC B
AB
BC
0
0
2
1
4
4
22 2
2
ºË°mËãÒÒÓÈ
∆
¯ÈmÓº}È}ÒmËãÒÒÓÈ
CA
+
ÓËäËÓ«Ë°«¹¯Òm©¹ºãÓ«Ë亮
ÏÈäËÓË°Ò°Ëä©}ºº¯ÒÓÈ
cÈÏËã 
sËãÒÓˮө˺­žË}ˆ©Óȹ㺰}º°ˆÒÒm¹¯º°ˆ¯ÈÓ°ˆmË



           jÏ‚°ãºmÒ« B ′ = 0 °ãË‚ˈˆº 2 B cos 2α − ( A − C ) sin 2α = 0 Òº}ºÓȈËã Óº
           
                                                          2B      1       2B
                                           tg 2α =           ; α = arctg     ¹¯Ò A > C 
                                                         A−C      2      A−C
                                                                                   
                                π
           ÒãÒÎË α =            ¹¯Ò A = C ˆºË°ˆ Ò°}ºä©®‚ºãÓÈ®ËÓ
                                4
      
      
      °º}ÈÎË䈺¹¯ÒˆÈ}º®ÏÈäËÓË}ºº¯ÒÓȈmËãÒÒÓÈ ∆ÓËÒÏäËÓ҈°«iË®°ˆmÒ
                                                                                                   A−C              1
           ˆËã Óº ÒÏ °ººˆÓº ËÓÒ® 1 + ctg 2 2α = 1 +                                    (             ) 2 = sin 2          Ò Ó˯ÈmËÓ°ˆmÈ
                                                                                                    2B                  2α
                           π
            0<α ≤            ¹ºã‚ÈËä
                           4
          
                                                              2B                                                A−C
                                    sin 2α =                                      ; cos 2α =                                     
                                                    4 B + ( A − C)
                                                         2                    2
                                                                                                         4 B + ( A − C) 2
                                                                                                            2

          
          
          sÈ®ËäˆË¹Ë¯ ÏÓÈËÓÒ« A ′ Ò C ′ jϹ¯Ë©‚Ò²°ººˆÓº ËÓÒ®ÒäËËä
          
          
                          1 + cos 2α                   1 − cos 2α A + C A − C
                  A′ = A              + B sin 2α + C              =         +         cos 2α + B sin 2α =
                               2                            2          2         2
                         A+C A−C                 A−C                       2B
                       =       +                               +B                      =                  
                          2         2      4 B 2 + ( A − C) 2       4 B 2 + ( A − C) 2
                           A+C 1
                       =      +   4 B 2 + ( A − C) 2                      .
                            2   2
                   
                   
                                                                   A+C 1
           kÓÈãºÒÓº¹ºã‚ÈË䈺 C ′ =                            −   4 B 2 + ( A − C ) 2 
                                                                    2   2
          
          
          |ˆ}‚È°ãË‚ˈˆº
          
                                               A′     0                           A+C               1
                                ∆ ′ = det                = A ′C ′ =           (            )   2
                                                                                                   − (4 B 2 + ( A − C) 2 ) =
                                                0     C′                           2                4
                                                                                                                                 
                                                      A B
                                     = AC − B 2 = det     =∆                           ,
                                                      B C
      
          ˆºË°ˆ mËãÒÒÓÈ ∆ ¯ÈmÓº}È}ÒmËãÒÒÓÈ A + C ÓËäËӫˈ°«¹¯Òm©¹ºãÓ«Ë亮
          ÏÈäËÓË°Ò°ˆËä©}ºº¯ÒÓȈ
               
              
              

Страницы