Методичка для геологов-заочников

1. :gZeblbq_kdZy ]_hf_ljby gZ iehkdhklb

JZkklhygb_ d f_`^m lhqdZfb

M1 (x1; y1 ) b M 2 (x2 ; y2 ) gZ

iehkdhklb

d = (x2 - x1 )2 + (y1 - y2 )2 = M1M 2

.

( 1 )

>_e_gb_ hlj_adZ \ ^Zgghf hlghr_gbb

>Zgu lhqdb M1 (x1; y1 ) b M 2

(x2 ; y2 ) =h\hjyl qlh lj_lvy lhqdZ

M (x; y ) e_`ZsZy gZ ^Zgghc ijyfhc ^_ebl hlj_ahd

M1M 2 \

hlghr_gbb λ _keb λ = ±

M1M

( λ

- iheh`bl_evgh _keb lhqdZ

MM 2

nhjfmeZf

x =

x1 + λx2

; y =

y1 + λy2

,

(λ ¹ -1).

( 2 )

1 + λ

1 + λ

Dhhj^bgZlu k_j_^bgu hlj_adZ hij_^_eyxlky ih nhjfmeZf

x =

x1 + x2

; y =

y1 + y2

.

( 3 )

2

2

\_jrbgZfb A(x1; y1 ),

B(x2 ; y2 ),

IehsZ^v lj_m]hevgbdZ k

C(x3 ; y3 ) gZoh^blky ih nhjfme_

S =

1

x

(y

- y

)+ x

(y

- y

)+ x

(y - y

)

.

( 4 )

2

1

2

3

2

3

1

3

1

2

- 3 -

H[s__ mjZ\g_gb_ ijyfhc

MjZ\g_gb_ \b^Z

Ax + By + C = 0,

( 5 )

]^_ A, B, C - ihklhyggu_ dhwnnbpb_glu A2 + B 2 ¹ 0 ;

x b y -

y = kx + b

( 6 )

hij_^_ey_lky ih nhjfme_

tgϕ =

k2 - k1

, k k

2

¹ -1.

( 7 )

1 + k1k2

1

Mkeh\b_ iZjZee_evghklb ijyfuo

k1 = k2 .

( 8 )

Mkeh\b_ i_ji_g^bdmeyjghklb ijyfuo

k = -

1

beb k k

= -1.

( 9 )

2

1

k2

1

MjZ\g_gb_ ijyfhc k m]eh\uf dhwnnbpb_glhf k ijhoh^ys_c

q_j_a ^Zggmx lhqdm M 0 (x0 ; y0 ) bf__l \b^

y - y0 = k (x - x0 ), k ¹ 0 .

( 10 )

y - y1

=

x - x1

; x

2

¹ x ; y

2

¹ y .

( 11 )

y2 - y1

x2 - x1

1

1

M]eh\hc dhwnnbpb_gl wlhc ijyfhc hij_^_ey_lky ih nhjfme_

k =

y2

- y1

.

x2

- x1

- 4 -

MjZ\g_gb_ ijyfhc \ hlj_adZo

MjZ\g_gb_ \b^Z

x + y = 1; a ¹ 0;b ¹ 0

( 12 )

GhjfZevgh_ mjZ\g_gb_ ijyfhc

MjZ\g_gb_ \b^Z

x cosα + y sinα - p = 0

( 13 )

gZau\Z_lky ghjfZevguf mjZ\g_gb_f ijyfhc A^_kv p

- ^ebgZ

fgh`bl_ev M =

1

.

±

A2 + B 2

>ey M gZ^h \aylv© ª_keb C < 0 agZd©-ª_keb C > 0 .

JZkklhygb_ hl lhqdb ^h ijyfhc

M 0 (x0 , y0 ) ^h ijyfhc

JZkklhygb_

d hl ^Zgghc lhqdb

Ax + By + C = 0 hij_^_ey_lky ih nhjfme_

d =

Ax0 + By0

+ C

( 14 )

.

A2 + B 2

(x − a)2 + (y − b)2 = R 2 ,

( 15 )

x2

+

y 2

= 1.

( 16 )

a 2

b2

- nhdmku weebikZ

A^_kv

a,b

-

ihemhkb weebikZ

F1 b F2

a 2 = c 2 + b2

Qbkeh

c

= ε < 1 lZd

dZd

a > c

gZau\Z_lky

a

wdkp_gljbkbl_lhf weebikZ

NhdZevgu_ jZ^bmku r1 = F1M b r2

= F2 M hij_^_eyxlky ih

nhjfmeZf

r1 = a + εx; r2 = a − εx.

=bi_j[heZ

M (x; y ) ,

=bi_j[heZ

– wlh

fgh`_kl\h

lhq_d iehkdhklb

F1M − F2 M

= 2a .

F1 b F2

- nhdmku

]bi_j[heu r1 = F1M b r2 = F2 M - nhdZevgu_

jZ^bmku

DZghgbq_kdh_ mjZ\g_gb_ ]bi_j[heu

x2

−

y 2

= 1.

( 17 )

a 2

b2

A^_kv a,b

- ihemhkb ]bi_j[heu ^_ckl\bl_evgZy

b fgbfZy

khhl\_lkl\_ggh

c 2

= a 2 + b2 .

- 6 -

x2

-

y 2

= 1

b

y 2

-

x2

= 1

a 2

b2

a 2

b2

gZau\Zxlky khijy`_ggufb

IZjZ[heZ

lhq_d iehkdhklb M (x; y),

IZjZ[heZ

– wlh

fgh`_kl\h

æ

p

ö

jZ\ghm^Ze_gguo hl ^Zgghc lhqdb F ç

,0

÷ gZau\Z_fhc nhdmkhf b

2

è

ø

h[hagZqZ_fh_ kbf\hehf

a11

a12

b hij_^_ey_fh_ jZ\_gkl\hf

a21

a22

a11

a12

= a a

22

- a

21

a .

( 1)

a21

a22

11

12

Hij_^_ebl_e_f lj_lv_]h ihjy^dZ

gZau\Z_lky qbkeh

hij_^_ey_fh_ jZ\_gkl\hf

a11

a12

a13

= a

a22

a23

-a

a21

a23

+a

a21

a22

.( 2 )

a

a

a

21

22

23

11

a

a

12

a

a

13

a

a

a31

a32

a33

32

33

31

33

31

32

Kbkl_fu ^\mo ebg_cguo mjZ\g_gbc k ^\mfy g_ba\_klgufb

Kbkl_fZ

ì a11x + a12y =b1

( 3 )

í

îa21x + a22y =b 2

b1

a12

a11

b1

x =

b2

a22

,

y =

a21

b2

,

( 4 )

a11

a12

a11

a12

a21

a22

a21

a22

]^_ D =

a11

a12

.

a21

a22

Kbkl_fZ ^\mo h^ghjh^guo ebg_cguo mjZ\g_gbc k lj_fy

g_ba\_klgufb

ìa11 x

+ a12 y + a13 z = 0

( 5 )

í

+ a22 y + a23 z = 0

îa21 x

bf__l j_r_gb_

x = t

a12

a13

, y = -t

a11

a13

,

z = t

a11

a12

,

a22

a23

a21

a23

a21

a22

]^_ t - ijhba\hevgh_ qbkeh

Kbkl_fZ lj_o h^ghjh^guo ebg_cguo mjZ\g_gbc k lj_fy

g_ba\_klgufb

ìa11 x + a12 y + a13 z = 0

ï

+ a22 y + a23 z = 0

ía21 x

ï

+ a32 y + a33 z = 0

îa31 x

a11

a12

a13

D =

a21

a22

a23

= 0.

( 7 )

a31

a32

a33

Kbkl_fZ lj_o ebg_cguo mjZ\g_gbc k ^\mfy g_ba\_klgufb

ì a11 x

+ a12 y

= b1

ï

+ a22 y

= b2

( 8 )

ía21 x

ïa

31

x

+ a

32

y

= b

î

3

a11

a12

b1

= 0 b kbkl_fZ g_ kh^_j`bl

kh\f_klgZ dh]^Z

a21

a22

b2

a31

a32

b3

x = r cosϕ

,

y = r sinϕ lh

z = r(cosϕ + i sin ϕ ).

( 12 )

>_ckl\by gZ^ dhfie_dkgufb qbkeZfb \ ljb]hghf_ljbq_kdhc

nhjf_

r1 (cosϕ1 + i sinϕ1 )× r2 (cosϕ 2 + i sinϕ 2 ) =

1)

,

( 13 )

r1 × r2 [cos(ϕ1 + ϕ 2 )+ i sin(ϕ1 + ϕ

2 )]

2)

r1

(cosϕ1 + i sinϕ1 )

=

r1

[cos(ϕ1 -ϕ 2 )+ i sin(ϕ1 -ϕ 2 )] , ( 14 )

r2

(cosϕ 2 + i sinϕ 2 )

r2

3)

[r(cosϕ + i sin ϕ )]n = r n (cos nϕ + i sin nϕ ) ,

( 15 )

4)Wk

= n r(cosϕ + i sinϕ ) =

n

æ

ϕ + 2kπ

+ i sin

ϕ + 2kπ ö

r çcos

n

÷,( 16 )

]^_ k = 0,1,2,..., (n -1).

è

n ø

3. FZl_fZlbq_kdbc ZgZeba

Ij_^_eu

Qbkeh

A gZau\Z_lky ij_^_ehf nmgdpbb f (x) ijb x ® a ,

?keb kms_kl\mxl dhg_qgu_ ij_^_eu lim f (x) b lim g(x) lh

x→a

x→a

1) lim[f (x) + g(x)]= lim f (x) + lim g(x) ,

( 1 )

x→a

x→a

x→a

- 10 -

2)

lim[f (x) × g(x)] = lim f (x) × lim g(x)

,

( 2 )

x→a

x→a

x→a

f (x)

lim f (x)

3)

lim

=

x→a

ijb lim g(x) ¹ 0 ) .

( 3 )

lim g(x)

x→a g(x)

x→a

x→a

Bkihevamy lZd`_ ke_^mxsb_ ij_^_eu

1)

lim

sinα

=1,

3) lim

ln(1 +α )

= 1 ,

( 4 )

α →0 α

α →0

α

lim(1 + α )

1

aα -1

2)

α

= e ,

4)

lim

= ln a

α →0

(1 +α )m -1

α →0

α

5) lim

= m .

α →0

α

lim

α

= m,α ~ α1 , β

~ β1 lh

x→a

β

α1

α

lim

= lim

= m .

( 5 )

x→a

β1

x→a

β

Ihe_agh

bkihevah\Zlv wd\b\Ze_glghklv

ke_^mxsbo

[_kdhg_qgh fZeuo _keb α → 0 lh

sinα ~ α , tgα ~ α , arcsinα ~ α , arctgα ~ α , ln(1 + α ) ~ α ,

aα -1 ~ α ln a,

(1 +α )m -1 ~ α × m.

( 5/ )

>bnn_j_gpbjh\Zgb_ nmgdpbc

Hij_^_e_gb_ Ijhba\h^ghc hl nmgdpbb y = f (x) \ lhqd_ x

gZau\Z_lky dhg_qguc ij_^_e

lim

f (x + Dx)- f (x)

= lim

Dy = f / (x) .

( 6 )

x→0

Dx

x→0

Dx