. , , ,

,,,

() . - : ()= + + ²+ +

(), ()=0, .. () , . . (), () .

3- .

. , .

. . ik . , ik=(-1) Mik .

∆ 3- : ∆=1111+1212+1313 .

. . . ¯¹ . . ¯¹= ¯¹ =.

. . , det≠0.

. . . . . . .: ¯¹=A/detA.

. . (") - .

. . . . . . n . ., . . . . . ., . (|E)"(E|A¯¹).

=  =

=¯¹  =¯¹

. m*n . S-, S-. (1≤S≤min(m,n)). ., . . . . . . . S. . S .

. . . . . . . ., . ., . . = 0.

. . . ,≠0.

=0, =0.

1.      R . . = R .

2.      R . . .

3.      . . R . . . . ,R . = r, .. . . . ≠0, =0, .

=(.), =(.), =(. .), Ấ=( . )

. .

|a11 a12 .. b1 .. a1m|

∆=|.| , ∆k=| a21 a22 .. b2 .. a2m|

|..|

| am1 am2 .. bm ..amm|

. . . . . 1=∆1/∆ , 2=∆2/∆

- ( )

. . . .

. . . || .

. . . . .

. ¯ .

. . .

- . . . . ., .

. α, β, γ . .

|r|=√(x²+y²+z²) x=|r|cosα y=|r|cosβ => cosα=x/√( x²+y²+z²)

e=(cosa,cosb,cosγ)

. .

n . . . a=α1*a1+α2*a2++αn*an x= α1*x1+α2*x2++αn*xn y=

X=(x1+ℓx2)/(1+ℓ) ℓ.

.

ab=|a||b|cos(ab) .. |b|cos φ= a b , |a|cosφ= b a , ab=|a| a b = |b| b a

: 1.() b=ba

2.() . . (αa)b=α(ab)

3. () . ⠠ a(b+c)=ab+ac

. . .

3 . . a,b,c . abc.

. c . . . b , b .

2- a b . [a*b] . . .:1)|[a*b]|=|a||b|sinα ;2)[a*b]┴a b;3) a b [a*b] , i jk.

. 1) | | = .

[a*b]=0 < = > a . b

: 1. [a*b]=-[a*b]

2. . . [(αa)*b]=α[a*b]

3. () . ⠠ [(a+b)c]=[a*c]+[b*c]

|i j k |

[a*b]=|x1 y1 z1|=|y1 z1|*i+

|x2 y2 z2| |y2 z2|

3 . a,b,c . a b . . . , . - .

V =. . . +, . abc .

abc=[ab]c=a[bc]

|x1 y1 |

abc=|x2 | < = > abc-.

|x3 | |x2-x1 y2-y1 |

V 3- . =mod|x3-x1 |

|x4-x1 |

.

a1,a2,an . . . , . α1,α2 αn, : α1*a1+α2*a2++αn*an=0

1. a1,a2,,an, n>1 < = > , . . . .

2. b . < = > .

3. 1 2 . ., , . . =*1+*2.

4. a,b,c . . < = > .

5. 1,2,3 ., . . =α1*1+α2*2+α3*3

6. . 4- . .

3- . .,.. d=x*e1+y*e2+z*e3 d(x,y,z) 123

߅

F(x,y)=0 -

F(ρ,φ)=0 . ρ, ρ= ρ(φ).

x=f(t)

y= φ (t) / - .

. - ρ= ρ(φ), - x= ρ(φ)*cos φ y= ρ(φ)*sin φ

. - Ax²+Cy²+Dx+Ey+F=0 (1)

. . . .

- (1) :

²/a²+y²/b²=1 . , . . . () =const,F1(-c,0), F2(c,0),c=√(a²+b²)

. . ξ=√(1-(b/a)²) . . x=a/ξ x=--a/ξ

²/a²+y²/b²=0 . . . . (0,0)

²/a²+y²/b²=-1 . . .

. *>0

²/a² -- y²/b²=1 --²/a² + y²/b²=1 . . | | .()=const

F1(-c,0), F2(c,0), c=√(a²+b²) , ξ=c/a, : =*b/a y=-- *b/a , : x=-a/ξ x=a/ξ |

. . /

²/a² -- y²/b²=0 / -

²=2px - . .

. . : ²=2px , x=-p/2 ,F(p/2,0) , r=x+p/2 |

oy : x²=2qy , y=-q/2 ,F(0,q/2) , r=y+q/2 |

y²=b² - || > -

y²=0 /

y²=--b² - . . .

=0, ≠0, (1) ²=2qy

. : = y=b

k=(y2-y1)/(x2-x1) , 1,1,, - . . | tg( / 2- ∩ )=(k2-k1)/(1+k1k2)

: y-y0=k(x-x0) | (++=0):

- : y-y0=-1/k*(x-x0) | tg( / 2- ∩ )=(A1*B2-A2*B1)/(A1*A2+B1*B2)

- (y-y1)/(y2-y1)=(x-x1)/(x2-x1) , (x2≠x1,y2≠y1) | || < = >A1/A2=B1/B2 , ┴ A1/B1=--B2/A2

- x=x1+(x2x1)*t y=y1=(y2y1)*t , t R

. 0(0,0) ++=0 : d=(A*x0+B*y0+C)/√(A²+B²)

- : (x-a)²+(y-b)²=R²

. - : Ax²+2Bxy+Cy²+Dx+Ey+F=0

α ctg2α=(A C)/2B

x=x cos α y sin α

y=x sin α +x cos α

-. b . lim y=f(x) x→a , ξ>0 . δ>0, x . . 0<|x-a|< δ, |f(x)-b|<ξ

() . - : ()= + + &sup2;+ + (), ()=0, .. () , . . (), () .

 

 

 

! , , , .
. , :