Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 97 стр.

UptoLike

Составители: 

Рубрика: 

cÈÏËã


sËãÒÓˮөËºË}©ÓÈ¹ãº°}º°ÒÒm¹¯º°¯ÈÓ°mË
j°¹ºãϺmÈÓÒË°Á˯Ò˰}º®°Ò°Ëä©}ºº¯
ÒÓÈ ÒÓºÈ ¹ºÏmºã«Ë ¹ºãÒ ºãËË
¹¯º°ºË ÈÓÈãÒÒ˰}ºË º¹Ò°ÈÓÒË ˺äË¯Ò
˰}Ò² ºË}ºm ºãÈÈÒ² ºËÓº®
°ÒääË¯ÒË® sȹ¯Òä˯ ¯ÈmÓËÓÒË °Á˯©
ËÒÓÒÓºº¯ÈÒ°È°ËÓ¯ºämÓÈÈãË}º
º¯ÒÓÈm°Á˯Ò˰}º®°Ò°ËäËËÒäË
ρ
=
1

nº¯äã© ¹Ë¯Ë²ºÈ äËÎ ¹¯«äº
ºãÓº® Ë}ȯºmº® °Ò°Ë亮 }ºº¯ÒÓÈ Ò
°Á˯Ò˰}º®ÒäË°ãËÒ®mÒ
x
y
z
=
=
=
ρ
ϕθ
ρ
ϕθ
ρ
θ
cos sin
sin sin
cos
z
M(
ρ
,
ϕ
,
θ
)



ρ



θ
O \

ϕ
P
[
èqxytvr
Òã«º¯ÈÓºº¹Ë¯Ë²ºÈ
ρ
ϕϕ
θ
=++ =
+
=
+
=
++
xyz
x
xy
y
xy
z
xyz
222
22 22
222
;cos ;sin ;
cos .
¡ÒãÒÓ¯Ò˰}È«°Ò°ËäÈ}ºº¯ÒÓÈ
{ ˲ °ãÈ«² È Ò°°ãËËä©®
¹¯º°¯ÈÓ°mËÓÓ©® ºË} ºãÈÈË º°Ëmº®
°ÒääË¯ÒË® äºÎË º}ÈÏÈ°« ºÓ©ä
¹¯ÒäËÓËÓÒË |qsqtléq·nxrvp xqxznu rvvé
lqtjz
ºãºÎËÓÒË º}Ò m ¹¯º°¯ÈÓ°mË m
ªº® °Ò°ËäË ºÓºÏÓÈÓº ÏÈÈË°« ¹¯Ò ¹º
äºÒ¯º®}ÒÒ°Ëã
{
ρ
,
ϕ
, h}
,¯Ò°Ë
ρ
=
QM 
ϕ
=∠
→→
(, )
Ox OP 
ºmãËmº¯«ÒË
º¯ÈÓÒËÓÒ«ä
ρ
0

ϕ

π

K ∈−+
(,)

z
 h Q

ρ
M(
ρ
,
ϕ
,z)
O

ϕ
 y

P
x
èqxytvr
cÈÏËã 
sËãÒÓˮө˺­žË}ˆ©Óȹ㺰}º°ˆÒÒm¹¯º°ˆ¯ÈÓ°ˆmË



           
    j°¹ºã ϺmÈÓÒ˰Á˯Ò˰}º®°Ò°ˆËä©}ºº¯                                          z
    ÒÓȈ ÒÓºÈ ¹ºÏmºã«Ëˆ ¹ºã‚҈  ­ºãËË                                       
    ¹¯º°ˆºË ÈÓÈã҈Ò˰}ºË º¹Ò°ÈÓÒË ˺äˈ¯Ò                                      
    ˰}Ò² º­žË}ˆºm º­ãÈÈ Ò² ˆºËÓº®                                         
    °Òääˈ¯ÒË® sȹ¯Òä˯ ‚¯ÈmÓËÓÒË °Á˯©                                        M(ρ,ϕ,θ )
    ËÒÓÒÓºº¯È҂°È°Ëӈ¯ºämÓÈÈãË}º                                        ρ
    º¯ÒÓȈ m °Á˯Ò˰}º® °Ò°ˆËäË ­‚ˈ Òäˈ                                   

    mÒ ρ = 1                                                                    θ
                                                                                                               O                                         \
             nº¯ä‚ã© ¹Ë¯Ë²ºÈ äË΂ ¹¯«äº                                        ϕP
    ‚ºã Óº® Ë}ȯˆºmº® °Ò°ˆË亮 }ºº¯ÒÓȈ Ò                                    
    °Á˯Ò˰}º®ÒäË ˆ°ãË‚ Ò®mÒ                                                
                           x = ρ cos ϕ sin θ                                        [
                                                                                    
                           y = ρ sin ϕ sin θ            
                                                                                     
                           z = ρ cosθ
                                                                                    èqxytvr
           
    Ò㫺­¯ÈˆÓºº¹Ë¯Ë²ºÈ
                                                                              
                                                                              
                                                         x                 y
                           ρ = x + y + z ; cos ϕ =           ; sin ϕ =
                                 2   2     2
                                                                                ;
                                                     x +y
                                                        2   2
                                                                         x +y
                                                                          2   2


                                                                                 
                                                 z
                                    cos θ =                           .
                                           x2 + y2 + z2
             
             
             
¡ÒãÒÓ¯Ò˰}È«°Ò°ˆËäÈ}ºº¯ÒÓȈ
             
             
            { ˆË² °ã‚È«² }ºÈ Ò°°ãË‚Ëä©®                                    z
    ¹¯º°ˆ¯ÈÓ°ˆmËÓÓ©® º­žË}ˆ º­ãÈÈˈ º°Ëmº®                                      
    °Òääˈ¯ÒË® äºÎˈ º}ÈÏȈ °« ‚º­Ó©ä
                                                                                      h Q
    ¹¯ÒäËÓËÓÒË |qsqtléq·nxrvp xqxznu€ rvvé                                       
    lqtjz                                                                          
            
                                                                                      ρM(ρ,ϕ ,z)
            ºãºÎËÓÒË ˆº}Ò m ¹¯º°ˆ¯ÈÓ°ˆmË m
                                                                                     
    ªˆº® °Ò°ˆËäË ºÓºÏÓÈÓº ÏÈÈˈ°« ¹¯Ò ¹º
                                                   O
    亝Òˆ¯º®}ÒÒ°Ëã {ρ, ϕ, h}, ¯Ò° Ë
         →           → →                           ϕ y
    ρ = QM  ϕ = ∠ (Ox , OP )  ‚ºmãˈmº¯« ÒË 
                                                                                     P
    º¯ÈÓÒËÓÒ«ä                                                                    
    
                                                                                     x
           ρ ≥ 0≤ϕπ K ∈ ( −∞,+∞ )                                      
                                                                                    
                                                                                    
                                                                                                            èqxytvr