CH1-4, Διαφορικη΄Γεωμετρία

1pi pipi

pi , pi, -

pi. pi pi,

pi pi pi pi pi, pi-

pi pi pi pi pi

pi pi .

1.1 pi;

pi pi, pi

, y2x D 1 ( pi!), , x2 C y2 D 1, pi, y x2 D 0:

y 2x = 1 y x2 = 0 x2 + y2 = 1

1.1.

pi pi

f .x; y/ D c;

1

2 1. pi pipi

pi f x y, c . pi pi

pi ,

C D f.x; y/ 2 R2 j f .x; y/ D cg: (1.1) pipi pi pi pipi R2, pi pi-

pi R3 pi, x R3

pi pi pi

y D 0; z D 0:, pi R3 pi pi pi

f1.x; y; z/ D c1; f2.x; y; z/ D c2: pi pi , pi . 1.1

pi .x; y/ pipi pi

pi f .x; y/ c.

pi pi pi pi, pi pi-

pi pi pipi: pi pi

pi . , .t/ -

t , pi pi pi

pi t ( R2 pi pipi pi,

R3 pi pi ).

pi pi Rn (

pipi n D 2 3, pi pipi).

1.1.1

pi pi Rn pi W .; / ! Rn, pi ; 1 < 1:

.; /

.; / D ft 2 R j < t < g: pi pi pi pi pi -

C pi ( ) C. pi

pi pi pi pi pi pi -

pi pi .

1.1.2

pi .t/ pi y D x2. .t/ D .1.t/; 2.t//, 1 2 pipi pi

2.t/ D 1.t/2 (1.2) t .; / pi ( pi-

pi ), pipi pi pipi

.1.t/; 2.t// pi t 2 .; /. , pi pi . 1.2: 1.t/ D t , 2.t/ D t2. pi pi pipi pi t pi pi ( .t/

1.1. pi; 3

pi t pi pi pi-

pi pi ), pi pipi pi .; /

.1;1/. pi pi pi :

W .1;1/! R2; .t/ D .t; t2/:

pi pi. pi

.t/ D .t3; t6/ ( .; / D .1;1/). .2t; 4t2/, pi- (pi) . pi pi pi

.

1.1.3

pipi pi x2 C y2 D 1. pi pi pi pi x D t pi pi pi, y D

p1 x2 ( pi pi pi y D

p1 x2). pi

pi pi

.t/ D .t;p1 t2/:

pi , p1 x2

pi 0. , pi y D p1 x2, pi

.

pi , pipi pipi-

. 1.t/ 2.t/

1.t/2 C 2.t/2 D 1 (1.3)

t 2 .; /, pi .1.t/; 2.t// pi t 2 .; /. pi pi . 1.3: 1.t/ D cos t , 2.t/ D sin t( cos2 t C sin2 t D 1 t). pi pi .; / D .1;1/, pi: pipi .; /

pi 2 .

pi pi pi pi pi pi

pi pi pi .

1.1.4

pi pi (pi )

.t/ D .cos3 t; sin3 t/; t 2 R: cos2 t C sin2 t D 1 t , x D cos3 t , y D sin3 t .t/ pi

x2=3 C y2=3 D 1: pi pipi pi .

. 1.1.5.

pi pi ( pi-

) pi pi . pi pi-

pi pi :

4 1. pi pipi

f W .; /! R pi dnfdt

pi n 1 t 2 .; /. f .t/ g.t/ , pi pipi pi f .t/C g.t/, f .t/g.t/, pi f .t/=g.t/ f .g.t// pi

.

pi .t/ (pi

1.1.1), pi :

.t/ D .1.t/; 2.t/; : : : ; n.t//;

d

dtDd1

dt;d2

dt; : : : ;

dn

dt

;

d 2

dt2Dd 21

dt2;d 22

dt2; : : : ;

d 2n

dt2

; ...

, d=dt P.t/, d 2=dt2 R.t/,pi. pi dn=dtn pi n 1 t 2 .; / , 1; : : : ; n .

pi pi pi

.

1.1.5

pi pi, pi pi P.t/ -pi .t/.

, pi

.t C t/ .t/t

pi pi .t/ .t C t/ C

( 1.2).

(t)

(t + t)

1.2. pi pi.

t , pi , -

pipi pi pi

C .t/.

.t C t/ .t/t

;

1.1. pi; 5

pi d=dt t . , pi

pipi pi pi d=dt

. , pi

C p C pi pi p pi

d=dt .

pi pi:

1.1.6

pi pi pi ,

pi ( ).

pi P.t/ D a t , pia , pi

.t/ DZ

d

dtdt D

Zadt D taC b;

pi b . a 0, pipi pi pi pi a pi b (

1.3).

(t )ta

b a

0

1.3.

a D 0, ( b). pi pi pi, pipi pi-

pi pi pi pi pi-

pi pi.

pi.

1.1.7

1 pi pi

.t/ D ..1C 2 cos t/ cos t; .1C 2 cos t/ sin t/; t 2 R( 1.4). ,

.t/ D 0 t D 2=3 t D 4=3. pi P.t/ D . sin t 2 sin 2t; cos t C 2 cos 2t/:

1... Limacon pi limax pi , .

6 1. pi pipi

,

P.2=3/ D .p3=2;3=2/; P.4=3/ D .

p3=2;3=2/:

pi pi pi ;

P.t/ t , pi t D 2=3 t D 4=3, pi 0 pi.

1.4. .

pi pi pi pipi pi -

pi pi pi : ,

pipi pi pi t pi,

.t/ 2 Rn. pi, 1.1.5 pipi pi : pi pi, pi

pi P.t/ pi pi t., pi, pi , pi pi

pi pi

pi pi 1.1.5.

pipi .

1.1.1 .t/ D .t2; t4/ pi pi y D x2;1.1.2 pi pi :

(i) y2 x2 D 1;(ii) x

2

4C y2

9D 1:

1.1.3 pi pi:

(i) .t/ D .cos2 t; sin2 t/;(ii) .t/ D .et ; t2/:

1.1.4 pi pi pi 1.1.3.

1.1. pi; 7

1.1.5 pi 1.1.4. pi

pi . pi -

pi ;

1.1.6

x2

p2C y

2

q2D 1;

pi p > q > 0 ( 1.5). Dq1 p2

q2,

.p; 0/ x , pi f1 f2. pi .t/ D .p cos t; q sin t/ pi . pi :

(i) pi p pi

f1 f2 pi p.

(ii) pi pi

p pi f1 f2 pi p.

(iii) p , pi f1 p pi f2 p

pi p.

f

pp

2f1

1.5.

1.1.7 pipi pi pi pi-

pi .

, x a > 0,

pi pi :

.t/ D a.t sin t; 1 cos t/:

1.1.8 1 .12; 0; 0/,

12 pi z.

.t/ D cos2 t 12; sin t cos t; sin t

pi pi pi -

8 1. pi pipi

1.6.

pi. pi pi

Viviani2 ( 1.6).

1.1.9 pi p pi

pi p pi p. pi-

pi.t/ D .2 cos tcos 2t; 2 sin tsin 2t/ pi t D =4.

1.2

pi v D .v1; : : : ; vn/ Rn,

kvk Dqv21 C C v2n:

u Rn, ku vk pi u v Rn.

pi pi pi pi-

, t pi , C .t/

.t C t/ , pi pi k.t C t/ .t/k:

pi t , ..t C t/ .t// =t pipi P.t/, pi pi

k P.t/kt: (1.4) pi ( )

C, pi , pi pi

t t , pi pi-

(1.4), pi. t

pi .

2... Vincenzo Viviani (16221703) .

Torricelli . pi 1.9 pi , Viviani

: pi , pi

pi pi .

1.2. 9

1.7. .

:

1.2.1

pi .t0/ s.t/

pi pi

s.t/ DZ tt0

k P.u/kdu:

pi, s.t0/ D 0 s.t/ t t0. pi .Qt0/, Qs pi s R Qt0

t0k P.u/kdu

Z tt0

k P.u/kdu DZ tQt0k P.u/kduC

Z Qt0t0

k P.u/kdu:

1.2.2

pi

.t/ D .ekt cos t; ekt sin t/;pi k ,

P.t/ D .ekt .k cos t sin t/; ekt .k sin t C cos t//;) k P.t/k2 D e2kt .k cos t sin t/2 C e2kt .k sin t C cos t/2 D .k2 C 1/e2kt :pi, .0/ D .1; 0/ ( pi-

)

s.t/ DZ t0

pk2 C 1ekudu D

pk2 C 1k

.ekt 1/:

. , s

10 1. pi pipi

1.8. pi.

pi .t0/,

ds

dtD d

dt

Z tt0

k P.u/kdu D k P.t/k: (1.5)

.t/ t ,

ds=dt ( pi

pi). .

1.2.3

W .; / ! Rn pi pi,

.t/ k P.t/k. pi P.t/ t 2 .; /.

pi pi pi pi

pi pi pi pi .

pi pi pi.

pi , pi .

pi a D .a1; : : : ; an/ b D .b1; : : : ; bn/ Rn

a b DnX

iD1aibi :

a b pi t , pi pi

d

dt.a b/ D da

dt bC a db

dt:

pi pipi pi pi

1.2. 11

d

dt.aibi / D dai

dtbi C ai dbi

dt:

1.2.4

n.t/ pi pi t ,

Pn.t/ n.t/ D 0 t , Pn.t/ n.t/ t .

, pi , R P .pi pi pi pi

n n D 1 pi t , piPn nC n Pn D 0;

2 Pn n D 0. pipi pi n D P .

1.2.1 pi .t/ D .t; cosh t/ - .0; 1/. pi

pi (. 2.2.4).3

1.2.2 pi :

(i) .t/ D13.1C t/3=2; 1

3.1 t/3=2; tp

2

;

(ii) .t/ D 45cos t; 1 sin t;3

5cos t

:

1.2.3 pipi pi pi

./ D .r cos ; r sin /;pi r ( .r; / pi

./). pi pi ;4

r./ pi . ,

, .

;

3... pi pi pi pi. pi ,

pi pipi pi, pi ,

. pi , , Jacob Bernoulli pi

pi y D a coshx, a D . coshx 1C x2

2pi pipi pi pi Taylor 2 coshx.

pi pi pi!4... . 1.3.3.

12 1. pi pipi

1.2.4 pi

pi pi . p q ,

pi pi pi , pi

.a/ D p, .b/ D b, pi a < b. , u ,

P u k Pk pi

.q p/ u Z ba

k Pkdt:

u D .q p/=kq pk, p q pi

kq pk.

1.3 pi

1.1.2 1.1.3 pi pi

pi pi.

.

1.3.1

pi pi Q W . Q ; Q/! Rn pi pipi W .; / ! Rn pi 11 pi pi W . Q ; Q/! .; / ( pi pi) pi- 1 W .; /! . Q ; Q/ pi

Q.Qt/ D ..Qt// Qt 2 . Q ; Q/: (1.6)

, pi , pi

Q:Q.1.t// D ..1.t/// D .t/ t 2 .; /:

pi pi pi ,

pi pipi .

1.3.2

1.1.3 pi x2 C y2 D 1

.t/ D .cos t; sin t/. pi

Q.t/ D .sin t; cos t/( sin2 t C cos2 t D 1). pi Q pi ,pipi pi pi

.cos.t/; sin .t// D .sin t; cos t/: .t/ D =2 t .

1.3. pi 13

pi pi 1.2, pi pi

. pi

pi pi pi .

1.3.3

.t/ pi pi

P.t/ 0 .t/ . pi 5 .

pi pi

, pi pi pi.

pi , pi pi

.

1.3.4

pi pi .

pi pi Q pi 1.3.1, t D .Qt/ D 1 Qt D .t/. - . .t// D t pi t pi pi

d

d Qtd

dtD 1:

d=d Qt pi . Q.Qt/ D ..Qt//, pi

d Qd Qt D

d

dt

d

d Qt ;

pi pi pipi d Q=d Qt pi d=dt pi.

1.3.5

.t/ pi, s (. 1.2.1),

, t .

pi ( ) s -

t

ds

dtD k P.t/k :

pi , pi pi-

pi pi,

.t/ D .u.t/; v.t//;5... pi regular point regular curve

. pi, singular point.

14 1. pi pipi

pi u v t ,

ds

dtDpPu2 C Pv2:

f .x/ D px .0;1/. , pi pi n 1

dnf

dxnD .1/n1 1 3 5 : : : .2n 1/

2nx.2nC1/=2:

u v t , Pu Pv, Pu2C Pv2. , Pu2C Pv2 > 0 t ,

ds

dtD f . Pu2 C Pv2/

t , pi s .

pi pi pi.

1.3.6

pi pi pi

.

pi pi pi pi W .; / ! Rn pi Q pi pi . t D .Qt/ Q.Qt / D .t/,

d Qd Qt D

d

dt

dt

d Qt ;

)

d Qd Qt

D

ddt

dtd Qt:

, kd Q=d Qtk D 1, pi d=dt pi .

, pi pi d=dt pi

. pi . 1.5 pi ds=dt > 0 t , pi s

pi,

1.3.5 s t . pi pi (

pi pi ) pipi s W .; /! R 11, . Q ; Q/, pis1 W . Q ; Q/! .; / . ( pi - pi pipi pipi pi pi

pi pi 1.5 pi

5.6.) D s1, Q pi

1.3. pi 15

Q.s/ D .t/ (. 1.6). pi

d Qds

ds

dtD d

dt;

)

d Qds

dsdt D

ddt

D dsdt ; ( . 1.5),

)

d Qds

D 1:

pi pi 1.3.6 pipi

pi pi.

1.3.7

pi Q pi :Q.u.t// D .t/ t;

pi u t . s (

),

u D s C c; (1.7)pi c . , u pi . 1.7 pi

c pipi pi, Q pi - .

pi pi pi pi pi

1.3.6 pipi u pi

du

dtD

ddt

D dsdt ( . 1.5);

pi u D s C c pi c. pi pi ,

pi pipi pi,

pi pi pi.

1.3.8

pi .t/ D .ekt cos t; ekt sin t/; 1.2.2 k Pk2 D .k2C1/e2kt : pi , . - .1; 0/ s Dpk2C1.ekt1/=k.6

t D 1kln

kspk2C1 C 1

, pi pi -

6... pi 1.2.2.

16 1. pi pipi

pi pi

Q.s/ D

kspk2 C 1 C 1

cos

1

kln

kspk2 C 1 C 1

;

kspk2 C 1 C 1

sin

1

kln

kspk2 C 1 C 1

:

1.3.9

7 pi

.t/ D .t; t2; t3/; t 2 R:

1.9. .

P.t/ D .1; 2t; 3t2/,

k P.t/k Dp1C 4t2 C 9t4:

pi , .

.0/ D 0

s DZ t0

p1C 4u2 C 9u4du:

pi pi pi

, . ( pi

pi .)

pi pi pi

pi.

7... pi twisted cubic.

1.3. pi 17

1.3.10

pi .t/ D .t; t2/ pi y D x2, P.t/ D .1; 2t/ pi pi , . Q.t/ D .t3; t6/ pi pi. PQ D .3t2; 6t3/, pi t D 0, pi Q .

1.3.1 pi pi pi ;

(i) .t/ D .cos2 t; sin2 t/ pi t 2 R.(ii) pi pi (i) 0 < t < =2.

(iii) .t/ D .t; cosh t/ pi t 2 R.

pi pi.

1.3.2 8 ( 1.10) pi pi pi -

.r; /

r D sin tan ; =2 < < =2: pi pi pi-

.t/ Dt2;

t3p1 t2

; 1 < t < 1;

pi .

1.10. .

8... (240180 pi..) pi .

pi pi ,

.

18 1. pi pipi

1.3.3 pi pi :

p , pi pi t0,

P.t0/ D 0 R.t0/ .t0/ - (, pipi

). :

(i) pi .t/ D .tm; tn/, pim n , .m; n/ D .2; 3/ .3; 2/.

(ii) 1.3.2

.

(iii) p, pi-

pi pi .

1.3.4 :

(i) Q pi pi , pi- Q .

(ii) Q pi O pi Q , O pi .

1.4 pi

pi pi pi , pi ,

pi , pi pi. ,

, pi pi pi , pi -

pi pi pi pi

pi . pi , pi

pi, pi pi . pi

pipi pi pi ( 1.11)

.t/ D .t2 1; t3 1/:

pi pi pi pi

pi ,

pipi pi. pi, pipi pi

pi .

1.4.1

W R ! Rn pi T 2 R. T -pi

.t C T / D .t/ t 2 R: T -pi pi T 0, .

1.4. pi 19

1.11. pi .

pi, T -pi, pi pi pi

pi T , pipi

. , pi 0-pi.

T -pi, pi pi pi pi

pipi jT j. , pi pi pi , W a; b ! Rn. pi pi a b,9 pi

pi pi .b a/-pi () pi W R ! Rn. pi, pi pi pi

.

pi pi T -pi .T /-pi,

.t T / D ..t T /C T D .t/:pi T -pi pi T 0, T -pi pi T > 0.

1.4.2

pi pi T pi

T -pi.

pi T pi (pi

pi ). pi

pi T pi .

9 pi a b pipi pi .

20 1. pi pipi

1.4.3

.t/ D .p cos t; q sin t/ ( 1.1.6) pi pi2 , 2-pi (pi pipi

pi ).10

pi, pi -

pi . pi pi

pi pi pi pi T

pi t pi pipi

T , pi 0 t T ,

`./ DZ T0

k P.t/kdt:

pi 1.3.6, pi

s DZ t0

k P.u/kdu

pi pi pi Q ( Q.s/ D .t//.

s.t C T / DZ tCT0

k P.u/kdu DZ T0

k P.u/kduCZ tCTT

k P.u/kdu D `./C s.t/;

, v D u T pi .u T / D .u/ ( pi P.u T / D P.u/), piZ tCT

T

k P.u/kdu DZ t0

k P.v/kdv D s.t/:

pi,

Q.s.t// D Q.s.t 0//() .t/ D .t 0/() t 0 t D kT () s.t 0/ s.t/ D k`./;pi k . pi pi Q pi pi `./. , Q , `./ pi Q . ,pi pi pi pi pi

.

pi pi 1.11, pi -

pi , pi

pi x < 0, pi -

. pi pi pi .

1.4.4

pi p pi pi

a b 10... T -pi pi pi .

1.4. pi 21

(i) .a/ D .b/ D p,

(ii) pi T , a b pipi T .

1.4.5

1.1.7 pi pi 2 . pi

, . ( pi

pi pi pi. 1.4.1 .)

1.4.1 Cayley

.t/ D .cos3 t cos 3t; cos3 t sin 3t/; t 2 R; pi . pi ;

( pi

pi 6 ).

1.4.2 pi pi pi

.

1.4.3 pi T1-pi T2-pi,

.k1T1 C k2T2/-pi pipi k1 k2.

1.4.4 W R ! Rn pi pi T0 - pi T0-pi. pi T -

pi T D kT0 pi k.

1.4.5 pi W R ! R T -pi pi T 0. pi pi T0 pi T0-pi.

pi pi pi pi pi .

pi pi pi, pi pi -

T0.

(i) pi T1; T2; T3; : : : T1 > T2 >

T3 > > 0 Tr -pi r 1.(ii) fTrg (i) pi pi

Tr ! 0 r !1.(iii) pi pi fTrg pi (i)

Tr ! 0 r !1 pi .

1.4.6 W R ! Rn pi pi T -pi piT > 0. .

22 1. pi pipi

1.5 pi pi

pi

pipi pi

pi pi pi .

pi , pi pi

pi pi. pi,

pi x2 C y2 D 0 . pi pipi pi f .x; y/ f .x; y/ D c, pi c , pi pi pipi pi

, pi pi pi pi

pi. pi pi c D 0 (pi f f c).

1.5.1

f .x; y/ (pi

pi f , , pi ). pi

, pi

C D f.x; y/ 2 R2 j f .x; y/ D 0g; @f=@x @f=@y . p C, -

.x0; y0/, pi pi pi .t/, -

pi pi 0, pi pi p t D 0 pi C t .

pi pi

pi ( pi pi pi

1.3.6). pi, pipi pi pi

. pi ( 5.6.2), -

pi pi pi-

pi.

pi pipi pi f

1.5.1, pi .x0 Cx; y0 Cy/ C p

f .x0 Cx; y0 Cy/ D 0: Taylor ,

f .x0 Cx; y0 Cy/ D f .x0; y0/Cx@f@x

Cy @f@y

;

pi pi pix y ( pi

pi .x0; y0/). pi,

x@f

@xCy @f

@yD 0: (1.8)

x y , .x;y/ pi C

1.5. pi pi pi 23

p, . 1.8 n D@f@x; @f@y

C

p ( 1.12).

(x, y)

p

n

C

x

y

1.12.

pi 1.5.1 n

C. pi pi @f@y ; p, n

pi x p, pi pi C p

pi y ( 1.13).

x

p

x0

y0

y

C

1.13.

x D x D x0 C .x; y/ p. ,

f .x; y/ D 0 (1.9) y y0 x x0.

24 1. pi pipi

pi pipi pi pi C p pi-

y ( @f=@y D 0), pi .. 1.14.y

p

xx0

C

1.14. pi pi y 0 p.

pi x D x D x0 C p, x D x0 C pi p.

f pi pi pi pi pi-

pi g.x/, x pi -

x0, y D g.x/ . 1.9 x0. pi pi pi C p

:

.t/ D .t; g.t//: g ( pi pipi pi

pi), P D .1; Pg/ pi pi. pi pi 1.5.1.

x2 + y2 = 1 x2 y2 = 1

1.15. pi.

pi pi pi pi pi -

pi 1.5.1. pi f .x; y/ pi

1.5. pi pi pi 25

pi pipi pi C pi pi

f .x; y/ D 0 . pi pi, C pi pi .

pi, x2Cy2 D 1 , pi x2y2 D 1 (. 1.15). pi f , pi pi-

pi pi C. pipi, C

, pi pi 11 C , pi-

pi ; pi C, ./ D ./ 11 .; /.

pi pi pi pi-

pi :

1.5.2

pi pi pipi .t0/ D .x0; y0/ , pi , pi f .x; y/,

x y pi pi x0 y0,

, pi pi 1.5.1 .t/

pi pi f .x; y/ D 0 t pi pi pi t0.

pi 1.5.2 pi pi

1.5.1.

.t/ D .u.t/; v.t//;pi u v . , pi

Pu.t0/ Pv.t0/ , Pu.t0/. u t pi t t0 ( 1.16).

u

u0

t0t

1.16.

pi pi 1.5.1, pi

t u D x0 x u.t/,

26 1. pi pipi

pi t t0. pi pipi h.x/, -

x pi pi x0, t D h.x/ u.t/ D x x x0 t t0. pi h .

f .x; y/ D y v.h.x// pi pi.

f .x; y/ pi -

pi 1.5.1 pi-

pi f .x; y/ D 0, pi pi 1.1.7. pi pi

pi pi f pi pi -

1.5.1 pi pi

.

1.5.1 pi C

y2 D x.1 x2/ . pi t pi C

.t/ D .t;pt t3/I

pi;

1.5.2 pi 1.5.1 pi R3

pi pi f .x; y; z/ D g.x; y; z/ D 0.1.5.3 pi pi 1.5.2 pi

R3 ( Rn). ( .)

pipi pi pi,

pi pi

pi pi ( pi) .

2 pi pi;

pi R3 -

, pi . pi pi pi

pi pi ( pi-

), pi pi pi

pipi ( pipi pi ). pi pi-

, pi pi pi.

2.1 pi

pipi pi pi -

pi. pipi pi pipi pi.

pipi pi, pi

pipi pi p pipi pi pi

pi p.

pi pi pi R2.

pi t t Ct , pi pi pi -pi .t/ pi ..t Ct/.t// n, pi n pi P.t/ .t/.

Taylor,

.t Ct/ D .t/C P.t/t C 12R.t/.t/2 C pipi; (2.1)

pi pi .pipi/=.t/2 t .

P n D 0, pi pi pi .t/

27

28 2. pi pi;

(t)

(t + t)

n

2.1. pi pipi pi pi pi .

1

2R.t/ n.t/2 C pipi:

, R pi- n. pi, pi pipi, pi-

pi pi

1

2k R.t/k.t/2:

pi :

2.1.1

pi pi t , pi .t/

.t/ pi k R.t/k. pi pi

Rn n 2. pipi, pi 1.1.6,

pi R D 0 pi, pipi pi.

pi pi pi pi

2.1.1 pi pi . -

R2 .x0; y0/ R, pi

:

.t/ Dx0 CR cos t

R; y0 CR sin t

R

:

2.1. pi 29

P.t/ D sin tR; cos t

R

,

k P.t/k Ds

sin tR

2Ccos

t

R

2D 1;

pi pi . pipi,

R.t/D 1R

cost

R; 1

Rsin

t

R

;

pi pi

k R.t/k Ds

1R

cost

R

2C 1R

sint

R

2D 1

R;

. pi

pi , pipi pi

pi.

pi .

pipi pi, , 1.3.6,

pi , Q , pi pi -pi pi Q . , pipi O pi , pi Q O . pi pi O pi- Q ( 1.3.4), pi, 1.3.7,

Q.t/ D O.u/;pi u D t C c c . , , pi

d Qdt

D d Odu

,

d 2 Qdt2

D ddu

ddu

D d

2 Odu2

;

pi Q O pi pi. pi pi ,

pi pipi pi (.

1.3.8 1.3.9). pi pi pi

pi pi .

2.1.2

.t/ pi R3, pi

D k R Pkk Pk3 ; (2.2)

pi ( ) d=dt .

30 2. pi pi;

, pi R2 pi pi pipi

() xy R3, . 2.2 pi pi pi pi

pi pipi pi.

pi s pi .

pi

P D ddt

D dds

ds

dt;

pi

D

d 2ds2

D

ddsd=dt

ds=dt

D

ddt

d=dtds=dt

ds=dt

D

dsdt

d2

dt2 d2s

dt2ddt

.ds=dt/3

: (2.3)pipi,

ds

dt

2D k Pk2 D P P ;

pi pi t pi

ds

dt

d 2s

dt2D P R :

pi . 2.3 pi

D

dsdt

2R d2s

dt2dsdtP

.ds=dt/4

Dk. P P/ R . P R/ Pk

k Pk4 :

pi

a .b c/ D .a c/b .a b/c(pi a;b; c 2 R3) pi

P . R P/ D . P P/ R . P R/ P:pipi, P R P , pi

k P . R P/k D k Pkk R Pk:pi,

k. P P/ R . P R/ Pkk Pk4 D

k P . R P/kk Pk4 D

k Pkk R Pkk Pk4 D

k R Pkk Pk3 :

pi . 2.2 P 0. pi pi pi.

2.1. pi 31

2.1.3

z pi

./ D .a cos ; a sin ; b/; 2 R;pi a b .

2.2. .

.x; y; z/ ,

x D a cos ; y D a sin ; z D b pi , x2 C y2 D a2, pi pi pi jaj z. jaj . 2 , .a cos ; a sin ; b/ pi-

pi z pi

2b. 2jbj . pi pi pi pi -

2.1.2. d=d , P./ D .a sin ; a cos ; b/,

k P./k Dpa2 C b2:

pi P./ pi , ( a D b D 0 pipi ). pi pi

pi 2.1.2: R D .a cos ; a sin ; 0/, R P D .ab sin ; ab cos ;a2/ pi

D k.ab sin ; ab cos ;a2/k

k.a sin ; a cos ; b/k3 D.a2b2 C a4/1=2.a2 C b2/3=2 D

jaja2 C b2 : (2.4)

pi .

32 2. pi pi;

pipi pi

pi . , pi b D 0 ( a 0). pipi pi pipi xy jaj,pi pi pi 2.1.2 pi

1=jaj. pi , . 2.4 pi

D jaja2 C 0 D

jaja2

D 1jaj :

pi pi a D 0 ( b 0). pipi z, , pi pi pi

. 2.4 pi pi a D 0.

2.1.1 pi pi pi pi:

(i) .t/ D13.1C t/3=2; 1

3.1 t/3=2; tp

2

.


5cos t

.

(iii) .t/ D .t; cosh t/.(iv) .t/ D .cos3 t; sin3 t/. (iv), pi 1 pi pi .1; 0/, .0;1/. pi pi 1.1.5.

2.1.2 , pi .t/ pi .t/ pi

> 0, .t/ t . pi

pi pi > 0.

2.2 pipi pi

pi pi pipi pipi

pi pi .

pi .s/ pi R2. -

d=ds ,

t D P pi t .pi

t pi pi-

ns pi pipi

pi t =2 ( 2.3).

1.2.4, Pt D R t pi ns .pi pi s

R D sns :

2.2. pipi pi 33

ns t

2.3. pi pi pipi pi.

s pi pi (pi ,

). , knsk D 1, D k Rk D ksnsk D js j; (2.5)

pi pi pi pi-

. 2.4 pi pi pi pi pi

pi ( pipi pi pi pi

s). s

.

s > 0

ns

ns

ns

ns

< 0 < 0 > 0

t

t

t

t

s s s

2.4. pi s .

.t/ , pi,

pi t, pi -

ns pi pi s

pi Q.s/, pi s . pi,

t D d=dtds=dt

D d=dtkd=dtk ;

34 2. pi pi;

ns pi t =2 ,

d t

dtD d t

ds

ds

dtD s

ds

dtns D s

ddt

ns :

pi pi pi -

pi pi . pi

, pi P.s/ pi '.s/ pi

P.s/ D .cos'.s/; sin '.s//: (2.6), '.s/ , pi pi pi

pi pi pipi 2 . pi

pi pi pi ':

2.2.1

W .; / ! R2 pi , s0 2 .; /, '0

P.s0/ D .cos'0; sin '0/: pipi pi ' W .; /! R '.s0/ D '0, pi . 2.6 s 2 .; /.

pi

P.s/ D .f .s/; g.s//:

f .s/2 C g.s/2 D 1 s; (2.7) .

'.s/ D '0 CZ ss0

.f Pg g Pf /dt:

'.s0/ D '0. pipi, f g , P' D f Pg g Pf '.

F D f cos' C g sin ' G D f sin' g cos':pi

PF D . Pf C g P'/ cos' C . Pg f P'/ sin ';pi

Pf C g P' D Pf .1 g2/C fg Pg D f .f Pf C g Pg/ D 0;pi pi . 2.7

pi

f Pf C g Pg D 0: pi pi Pg f P' D 0. pi PF D 0 F . pi G .

F.s0/ D f .s0/ cos'0 C g.s0/ sin '0 D cos2 '0 C sin2 '0 D 1;

2.2. pipi pi 35

pi pipi G.s0/ D 0. pi f cos' C g sin' D 1; f sin ' g cos' D 0

s. pi pi f D cos', g D sin ', ' pi . 2.6.

,

.s0/ D '0 P.s/ D .cos .s/; sin .s// s 2 .; /, pi n.s/

.s/ '.s/ D 2n.s/ s 2 .; /:pi ' , n s.

n . , pi n.s0/ n.s1/ pi s1 2 .; /, pi, , n.s/ pipi pi n.s0/ n.s1/

s s0 s1. pi pi

n.s0/ n.s1/ ! pi, pi n

pi s, .s0/ D '.s0/ D '0, pipi n D 0 .s/ D '.s/ s 2 .; /.

2.2.2

' 2.2.1 pi

pi '.s0/ D '0. pi pi

pi pi pi .

2.2.3

.s/ pipi pi '.s/ pi

,

s D d'ds

:

pi, pi pi pi pi

pi. pi 2.4, pi pi

t pi

pi pi

s.

pi . 2.6 pi t D .cos'; sin '/,

Pt D P'. sin '; cos'/:pi ns D . sin '; cos'/, pi pipi pi Pt D sns .

36 2. pi pi;

2.2.4

pi pi pi ( 1.2.1).

pi pi .t/ D .t; cosh t/ pi P D .1; sinh t/

s DZ t0

p1C sinh2 tdt D sinh t:

pi, ' pi P x, tan' D sinh t D s;

) sec2'd'

dsD 1;

) s D d'ds

D 1sec2'

D 11C tan2 ' D

1

1C s2 :

2.2.3 pi pi pi

pi pi `, -

Z `0

s.s/ds: (2.8)

2.2.5

pi pi pipi pi

pipi 2 .

pi pipi pi `

. 2.2.3 pi pi

Z `0

d'

dsds D '.`/ '.0/;

pi ' pi . `-pi (. 1.4):

.s C `/ D .s/: pi

P.s C `/ D P.s/; P.`/ D P.0/. pi, . 2.6,

.cos'.`/; sin'.`// D .cos'.0/; sin'.0//;pi pi '.`/ '.0/ pipi 2 .

pipi , pi pi

pipi pi ,

2.2. pipi pi 37

pi pi.

R2 R2 piM W R2 ! R2 M D Ta ;

pi pi pi

,

.x; y/ D .x cos y sin ; x sin C y cos /; Ta a,

Ta.v/ D vC a; pipi .x; y/ v R2 (. 1).

2.2.6

k W .; /! R , pi pi W .; /! R2 pi pi pi k.

pipi, Q W .; / ! R2 pi pi pi pi k, pi M

R2

Q.s/ D M..s//; s 2 .; /:

pi pi pi s0 2 .; / s 2 .; /

'.s/ DZ ss0

k.u/du; (pi. 2.2.3),

.s/ DZ s

s0

cos'.t/dt;

Z ss0

sin '.t/dt

:

pi pi

P.s/ D .cos'.s/; sin '.s//; pi pi ' x. pi,

, 2.2.3, pi

pi

d'

dsD d

ds

Z ss0

k.u/du D k.s/:

, Q'.s/ pi Q . pi,PQ.s/ D .cos Q'.s/; sin Q'.s//;

) Q.s/ DZ s

s0

cos Q'.t/dt;Z ss0

sin Q'.t/dtC Q.s0/: (2.9)

2.2.3 k.s/ D d Q'=ds,

Q'.s/ DZ ss0

k.u/duC Q'.s0/:

38 2. pi pi;

pipi . 2.9 a

Q.s0/ Q'.s0/, pi

Q.s/ D TaZ s

s0

cos.'.t/C /dt;Z ss0

sin.'.t/C /dt

D Tacos

Z ss0

cos'.t/dt sin Z ss0

sin'.t/dt;

sin

Z ss0

cos'.t/dt C cos Z ss0

sin '.t/dt

D TaZ s

s0

cos'.t/dt;

Z ss0

sin '.t/dt

D Ta ..s//:

2.2.7

pipi pi pi pi

. pi pi pi

pi s pi pi , pi .

2.5

s D :A priori, pi s D pi pi s D pi , pi s

s (. 2.2.2). , spi pipi pi .pi, s D pi s D pi. , s .

pi pipi , pi s, pi-

pi pi pi pi

s . , pi pi pi-

pi s pi pi

. pi pi

pi , pi pi pi pi

pi ( ).

pi

R

.s/ DR cos

s

R;R sin

s

R

:

pi

t D P.s/ D sin s

R; cos

s

R

;

pi =2 C s=R x( 2.5).

2.2. pipi pi 39

t

s = R

xs = R

2.5. pi R.

pi pi

d

ds

2C s

R

D 1

R:

pi, s > 0, 1=s pi pi s .

s < 0, pi pi

Q.s/ DR cos

s

R;R sin s

R

( pi pi pi -

R) pi pi 1=R. pi, R D 1=spi pi pi pi s.

pi pipi pi pi

pi ( pi pi) pi

2.1.1.

2.2.8

2.2.6, pi pipi pi pi

pi pi pipi .

pi pi pi pipi pi. pi-

, pi pi s.s/ D s. pi 2.2.6 pi s0 D 0, '.s/ D

R s0udu D s2

2,

.s/ DZ s

0

cos

t2

2

dt;

Z s0

sin

t2

2

dt

:

pi -

. (

, pi Fresnel,1 pi pi

1... Augustin-Jean Fresnel (17881827)

40 2. pi pi;

2.6. pi Cornu.

Cornu,2 pi pi Euler). 2.6

pi .

2.2.6 -

pi pi pi. pi

( ) pi k 0, pipipi pi pi pi k

pi k. 2.2.6 pi

. pi pi () pi pi pipi

y

x

y

x

2.7. 2.2.6 pi.

pi. Fresnel pi , pi,

pi Fresnel pi pi .2... Marie Alfred Cornu (18411902) .

2.2. pipi pi 41

x ( ) 1 x 1 pipi - pipi. ( pi pi pi pi

pi pi 9.4.3). ,

pi pi x 0 pi x. pi pi , pi pi -

. . 2.2.3 2.2.6 pi

pi pi pi.

2.2.1 , pipi pi ,

Pns D st:

2.2.2 pi pi pipi pi

.t/ t . ( -

2.1.2.)

2.2.3 Q pipi pi. , Q pipi pi

M R2, pi pi-

s Qs Q M , Qs D s M (, Q pi). , , Q pipi, Q pipi pi R2.

2.2.4 k pi pi pipi pi C

. , Ca C

v 7! av (pi a ), pipi Ca s 1aksa

.

pi pi pi

pipi pi C. , s C

pi , ' pi pi C

, T pi,

T cos' D ; T sin' D s;pi ; (pi

). pipi C

s D 1a

1C s

2

a2

1;

pi a D =, pi C pi pi pi 2.2.4

pipi.

42 2. pi pi;

2.2.5 .t/ pipi pi . pi

pi pi

.t/ D .t/C ns.t/: , s.t/ 1 t , -pi pi pi s=j1 s j.

2.2.6 pi pi pipi pi-

.s0/ -

pi . pi pi

pi .

, pi

pi .s0/ .s0 s/ pi

.s0/ D .s0/C 1s.s0/

ns.s0/

t . C .s0/ pi pi

.s0/ 3 .s0/, .s0/

pi .s0/. C, pi

pi .s0/, 1=js.s0/j D 1=.s0/, pi pi .

2.2.7 pi pi ,

pi pi, pi

( pipi pi,

pi ). pi

Ps.s/ 0 s ( d=ds), Ps > 0 s ( pi, pi pi

s pi s). 1s.s/

( pi-

pi ) pi pi pi

. pi pi

( pi

).

.t/ D a.1 sin t; 1 cos t/; 0 < t < 2;pi a > 0 ,

.t/ D a.t C sin t;1C cos t/(. 1.1.7) , pi, pi pi-

pi pipi.

2.2.8 ` pi .0/ pipi pi -

.s/. pi pi

pi pi , pi

.s/ D .s/C .` s/ P.s/;3... pi osculating circle.

2.3. pi 43

pi 0 < s < ` d=ds. pi

( pipi pi, -

pi

). pi pi pi s pi

, s.s/ > 0 s. pi pi

1=.` s/.

2.2.9

.t/ D .t; cosh t/ l D 0 (. pi ) 4

x D cosh11

y

p1 y2:

. 8.3 pi pi .

2.2.10 pipi pi .s/

pi ` pi pi a,

pi ` p D .0/. q pi . .s/ pi q pi-

pi pi ` s (

). .s/ pi

a pi P . .s/ D p C saC .s/.q .s//;

pi ' pi pi '.

pipi P .s/ .s/.q .s// D 0:

pi

pi pi pi pi `. .

1.1.7 pipi.

2.3 pi

pi ( pi) R3, -

pi . pipi pi pi

pi (. 2.2.6), pi .

pi, 1 pipixy a D b D 1=2(. 2.1.3) pi 1 pi, pi

pi pi pi pi

pi R3. pi pi

, pi , pi pi

pi pi R3.

4... tractrix pi trahere pi , .

44 2. pi pi;

.s/ pi R3 t D P pi . pi .s/ ,

.s/

n.s/ D 1.s/

Pt.s/: (2.10)

kPtk D , n . pipi, 1.2.4 t Pt D 0, t n .pi

b D t n (2.11) , t n. b.s/

.s/. pi ft;n;bg 5 R3, pi ,

b D t n; n D b t; t D n b:

n

b

t

2.8. Frenet.

b.s/ s, Pb b. pipi pi pi -

u v pi s:

d

ds.u v/ D du

ds vC u dv

ds:

( pi pi pi -

pi pi -

). pi b D t n piPb D Pt nC t Pn D t Pn; (2.12)

pi (2.10) n Ptn D nn D 0:pi . 2.12 pipi Pb t. t b, Pb pipi pi n,

Pb D n; (2.13)5... Frenet.

2.3. pi 45

pi , pi ( pi-

pi pi

). pi

.

, pi

pi . pi pipi pi-

, pi , pipi pi pi-

pi pi

u D s C c;pi c . pi pi

pipi pi pipi:

t 7! t; Pt 7! Pt; n 7! n; b 7! b; Pb 7! Pb:pi . 2.13 pi 7! . pi, pi pi.

pi pi 2.1.2,

pi pi

pi :

2.3.1

.t/ pi R3 pi pi,

pi pi

D . P R/ k P Rk2 ; (2.14)

pi d=dt .

, pi, .t/

.t/ pi pi pi .t/ ,

2.1.2 pi pipi

pi 2.14.

pi pi pi . 2.14 pi pi

pi 2.1.2. pi

pi pi, pi pi . 2.14

pipi pi.

pipi pi . -

pi . 2.11 2.13 pi

D n Pb D n .t n/PD n .Pt nC t Pn/ D n .t Pn/: n D 1

Pt D 1

R , pi

D 1R P d

dt

1

R

D 1R P

1

P

2R

D 12 . P R/;

R . P R/ D 0 R . P / D . P R/. pipi . 2.14, , , P R , pi

k P Rk D k Pkk Rk D k Rk D :

46 2. pi pi;

pipi, s pi ,

d

dtD ds

dt

d

ds;d 2

dt2Dds

dt

2d 2

ds2C d

2s

dt2d

ds;

d 3

dt3Dds

dt

3d 3

ds3C 3ds

dt

d 2s

dt3d 2

ds2C d

3s

dt3d

ds:

pi,

P R Dds

dt

3 d

ds d

2

ds2

;

. P R/ Dds

dt

6 d 3

ds3d

ds d

2

ds2

:

D

d3

ds3dds d2

ds2

dds d2ds2

2 D

. P R/k P Rk2 :

2.3.2

pi ./ D .a cos ; a sin ; b/ pi - 2.1.3.

P./ D .a sin ; a cos ; b/; R./ D .a cos ;a sin ; 0/;./ D .a sin ;a cos ; 0/; P R D .ab sin ;ab cos ; a2/;

k P Rk2 D a2.a2 C b2/; . P R/ D a2b;

D . P R/ k P Rk2 Da2b

a2.a2 C b2/ Db

a2 C b2 :

2.3.2 -

b D 0 pipi pi pipixy. , pi

pi pi pi.

2.3.3

pi R3 pi pi (

), pi pipi

pi.

2.3. pi 47

pi pi pi ( pi

pi pi , pi

pi pi pipi). pi-

s d=ds , .

pi pi pipi v N D d , piN , d v 2 R3. pi pi N . N D d pis pi

t N D 0; (2.15)) Pt N D 0 ( PN D 0);) n N D 0 ( Pt D n);) n N D 0 ( 0): (2.16)

pi 2.15 2.16 pipi t n N . pi

b D t n pi N . N b b.s/ ( ) s, pipi

b.s/ D N s b.s/ D N s. pipi b . pi Pb D 0, D 0.

, pi D 0 pi. pi . 2.13 pipi Pb D 0, b . pi pi pi pipipi pi pipi v b D . pi

d

ds. b/ D P b D t b D 0;

pi pi pipi b () , d . pi pi pipi v b D d .

pi pi pi . -

, pi

Pt D n Pb D n(pi pi n ),

pi Pn. . ft;n;bg - R3,

t n D b; n b D t; b t D n:

Pn D Pb t C b Pt D n t C b n D t C b: pipi :

2.3.4

pi R3 pi pi-

48 2. pi pi;

,

Pt D nPn D t C b (2.17)Pb D n:

2.17 Frenet-Serret. pi0@ 0 0 0 0 0

1A pi Pt, Pn Pb t, n b -, .

pi . ( pi -

2.3.6.)

pi Frenet-Serret:

2.3.5

pi R3 pi

, pi ( ).

pi pi pi pi -

2.2.7 2.3.3, pi pi pi -

pi pi,

pipi pi .

pi 2.3.3, b -

pi pipi, , b.

d

ds

C 1

n

D t C 1

Pn D 0;

pi pi pi

Frenet-Serret

Pn D t C b D t; ( D 0)( pi C 1

n 2.2.6). pi,

C 1n , a,

k ak D

1 n

D 1 : pi , S, a

1=. S , C, pi

pi C. r C, D 1=r , pi r D 1= pi S. pi, C S, pi a S. a C

v b D a b.

2.3. pi 49

2.2.6 -

pi .

2.3.6

.s/ Q.s/ pi R3 pi- .s/ > 0 .s/ s, pi M

R3

Q.s/ DM..s// s:pipi, k t k > 0 pi, pi pi

R3 pi pi k t .

pi t;n b pi ,

, Qt; Qn Qb Q . s0 pi s, t.s0/ Qt.s0/, pi pi pi pi

, .t.s0// D Qt.s0/. On D .n.s0// Ob D .b.s0//. pi ' On Qn.s0/. 0 pi ' pi pi pi

pi Qt.s0/, 0 Qt.s0/ pi On Qn.s0/. pi ft.s0/;n.s0/;b.s0/g fQt.s0/; Qn.s0/; Qb.s0/g - R3, 0 pi t.s0/;n.s0/ b.s0/ Qt.s0/; Qn.s0/ Qb.s0/, . , M M D T Q.s0/.s0/ 0 . 2.3.5, pi DM./ , T , N B pi

, ,

.s0/ D Q.s0/; T .s0/ D Qt.s0/; N .s0/ D Qn.s0/; B.s0/ D Qb.s0/: (2.18) pipi pi:

A.s/ D Qt T C Qn N C Qb B: . 2.18 A.s0/ D 3. pi , Qt T , Qt T 1, Qt D T . Qn N Qb B. pi A.s/ 3, Qt D T , Qn D N Qb D B. , pi pi A , pi Qt D T , PQ D P , pi Q.s/ .s/ . pi 2.18, pipi , Q D .

pi, pi pi A

. pi Frenet-Serret pi

PA D PQt T C PQn N C PQb B C Qt PT C Qn PN C Qb PBD Qn T C . Qt C Qb/ N C . Qn/ B C Qt NCQn .T C B/C b .N /;

pi .

50 2. pi pi;

, pi pi

pi

PT D kN ; (2.19)PN D kT C tB; (2.20)PB D tN (2.21)

T .s/;N .s/;B.s/ T .s0/;N .s0/;B.s0/

i D .1; 0; 0/, j D .0; 1; 0/, k D .0; 0; 1/, . pi 0@ 0 k 0k 0 t

0 t 0

1A pi PT ; PN PB T ; N B -, T ; N B s

(. 2.3.6).

.s/ DZ ss0

T .u/du:

P D T , , T , -. pi, PT D kN . 2.19, , N , k pi N . ,

B T N , B D T N , pi s pi 1 s. k D i j , .s0/ D 1, .s/ D 1 s. B

, t , . 2.21, .

2.3.1 pi ; ; t; n b pi pi pi

pi pi Frenet-Serret:

(i) .t/ D13.1C t/3=2; 1

3.1 t/3=2; tp

2

.


5cos t

.

pi (ii) ,

pipi pi .

2.3.2 pi R3 pi pi > 0

.

2.3.3 pi R3 pi > 0

pi -

a. pi

pi D cot . ,

2.3. pi 51

pi pi pi D , pi , pi .

pipipi, 2.1.3 2.3.2 -

. pi pi

.

2.3.4 .t/ pi .t/ > 0 .t/ 0 t . , , pi

pi ,

D d

ds

P2

: (2.22)

, . 2.22

2 C . P/2 D r2

pi () r , pi D 1= D 1= , pi pi r . pi . 2.22

pi Viviani ( 1.1.8).

2.3.5 P 33 pi a 2 R3, M.v/ D PvCa R3 (. 1). -

pi R3, pi DM./ pi - . pi , t, n, b T , N , B pi-

, ,

T D P t, N D Pn B D Pb.2.3.6 .aij / 3 3 pi ( aij D aj i

i; j ). v1; v2 v3 pi s pi pi

Pvi D3X

jD1aijvj ;

i D 1; 2 3. pi pi pi s0 - v1.s0/; v2.s0/ v3.s0/ .

v1.s/; v2.s/ v3.s/ s.

pipi

pi pi .

3 pi

pi pi pi,

pi pi pi

, pi pi.1 pi -

pi, pi pi pi pi

pi pi , pi

pi. pi -

pi, pi pi-

pi pi pi

.

3.1 pi pi

pipi pi :

3.1.1

pi pi R2 pi R2 .

pi, pi pi R2, pi

pi Jordan, pi pi

pipi : , pi

( R2 pi )

pi R2, pi int./ ext./,

:

1... pi pi pi

global.

53

54 3. pi

(i) int./ , pi -

.

(ii) ext./ .

(iii) int./ ext./ ,2 -

: pipi pi pi

pi pi ( pi pi

int./ ext./ pipi pi ).

3.1.2

.t/ D .p cos t; q sin t/, pi p q , pi pi pi 2 . -

, , n.x; y/ 2 R2 j x2

p2C y2

q2< 1

o

n.x; y/ 2 R2 j x2

p2C y2

q2> 1

o.

pi

pi pi pi pi. .. p

pi pi pi pi

3.1;

p

3.1. p pi;

3.1.3

1.1.7 pi pi,

. 3.1.1.

pi pi pi-

pi pi .

pi pi ns

2... pi .

: pi R2 ( Rn )

pi ( ) pi . pi

. .

pi pi pi.

3.1. pi pi 55

(. 2.2) pi int./ . pi pi

pi pi t t pi. 3.2 pi pi.

pi pi pi 3.1;

t

ns

ns

t

3.2. pi pi pi.

pi pi pi-

.

3.1.4 ( pi Hopf3)

pi pi pi pi R2 2 . pi 3.1.4 pi pi

pi pi . pi

( pi) 13.1.

2.2.5 pi pi-

pi R2 pipi 2 .

Hopf pi pi ,

pipi 1. pi pi 2.2.5 pi - Hopf ' pi pi

2 pi - pi pi 2 . pi pi pi 3.1,

pi .

3.1.1

.t/ D ..1C a cos t/ cos t; .1C a cos t/ sin t/ ;pi a , pi pi jaj < 1, jaj > 1 pi pi R2, pi pi .

a D 1;3... Umlaufsatz pi

pi. Heinz Hopf (18941971) pi pi.

56 3. pi

3.2 pi

pi pi pi pi

A./ D

int./

dxdy: (3.1)

pi pi pi .

Green. f .x; y/ g.x; y/ (

pi ), pi

pi pi, int./

@g

@x @f@y

dxdy D

Z

f .x; y/dx C g.x; y/dy:

pi Green pi pi -

pi .

3.2.1

.t/ D .x.t/; y.t// pi pi pi R2 pi T ,

A./ D 12

Z T0

.x Py y Px/dt: (3.2)

pi f D 12y g D 1

2x Green,

A./ D 12

Z

xdy ydx;

pi pi pipi . 3.2.

, pi pi . 3.2 pi pi t , pi

3.1.1 A./ pi pi-

.

pi pi pipi pi pi-

.

3.2.2 (pi )

pi pi, `./ A./ pi

pi pi,

A./ 14

`./2;

.

pi , pi-

pi `./ D 2R A./ D R2, pi R . pi pi pi :

3.2. pi 57

3.2.3 ( Wirtinger)

F W 0; ! R F.0/ D F./ D 0, Z 0

dF

dt

2dt

Z 0

F.t/2dt;

F.t/ D D sin t t 2 0; , piD .

pi pi Wirtinger, pi pi

pi pipi pi .

pi pi pi pi pipi-

pi. pi, pi, pi

pi pi s. , pi

3.2.2, pi pi . -

pi pi s

t D s`./

; (3.3)

pipi pi pi pi pi , s

`./, t . pi, pi

pi pi t . 3.3.

pipi, pi `./ A./ pi -

.t/ 7! .t/ C b, pi b (. 3.2.1). b D .0/, pi pipi .0/ D 0, .

pi 3.2.2 pi `./ A./ -

pi pi :

x D r cos ; y D r sin :pi pi

Px2 C Py2 D Pr2 C r2 P2; x Py y Px D r2 P;pi d=dt . pi, pi . 3.3 pipi

Pr2 C r2 P2 Ddx

dt

2Cdy

dt

2D

dx

ds

2Cdy

ds

2!ds

dt

2D `./

2

2; (3.4)

pi .dx=ds/2 C .dy=ds/2 D 1. pipi, pi . 3.2

A./ D 12

Z 0

.x Py y Px/dt D 12

Z 0

r2 Pdt: (3.5)

pi 3.2.2 pipi

`./2

4A./ 0;

58 3. pi

. pi . 3.4 pipi Z 0

.Pr2 C r2 P2/dt D `./2

:

, pi . 3.5, pi

`./2

4A./ D 1

4

Z 0

.Pr2 C r2 P2/dt 12

Z 0

r2 Pdt D 14I;

pi

I DZ 0

.Pr2 C r2 P2 2r2 P/dt: (3.6)

pi, pi 3.2.2, pipi I 0, I D 0 .

pi pi pipi

I DZ 0

r2. P 1/2dt CZ 0

.Pr2 r2/dt: (3.7)

pi . 3.7 pi 0, 0 Wirtinger (pi F D r : pi- r.0/ D r./ D 0 pi .0/ D ./ D 0). pi I 0. pi, . 3.7 0, I . pi -

P D 1 t , r D D sin t pi D (pi Wirtinger). D t C, pi , pi r D D sin. /. pi pi pi D, pi

pi 3.2.2 (. 3.3).

r

3.3. D: r D D sin. /.

pi pi Wirtinger.

3.2. pi 59

G.t/ D F.t/= sin t . d=dt , Z 0

PF 2dt DZ 0

. PG sin t CG cos t/2dt

DZ 0

PG2 sin2 tdt C 2Z 0

G PG sin t cos tdt CZ 0

G2 cos2 tdt:

4

2

Z 0

G PG sin t cos tdt D G2 sin t cos t j0 Z 0

G2.cos2 t sin2 t/dt

DZ 0

G2.sin2 t cos2 t/dt;

Z 0

PF 2dt DZ 0

PG2 sin2 t CZ 0

G2.sin2 t cos2 t/dt CZ 0

G2 cos2 tdt

DZ 0

.G2 C PG2/ sin2 tdt DZ 0

F 2dt CZ 0

PG2 sin tdt;

pi Z 0

PF 2dt Z 0

F 2dt DZ 0

PG2 sin tdt:

pi 0 PG D 0 t , G.t/ , D, t , pi pi F.t/ D D sin t .

3.2.1 `./ A./ pi

.

3.2.2 pi

x2

p2C y

2

q2D 1

(pi p q ), pi Z 0

qp2 sin2 t C q2 cos2 tdt 2ppq;

p D q.4 pi G pi ( pi

G.t/2 sin t cos t pi ). G.t/ t D 0 , F .t/= sin t 0=0 . pipi G pi pi pi. pi lHospital.

60 3. pi

3.3

pi pi pi -

pipi pi.5 pi pi

int./ , pi

pipi int./ pi int./.

3.4. pi.

3.3.1

pi .t/ R2 pi -

pi s, pi ds=dt D 0.pi pi pi,

pi .

3.3.2

pi pi pi, pi pi

.t/ D .p cos t; q sin t/, pi p q ,

s.t/ D pq.p2 sin2 t C q2 cos2 t/3=2 :

pi

ds

dtD 3pq.q

2 p2/ sin t cos t.p2 sin2 t C q2 cos2 t/5=2

, pi

t D 0; =2; 3=2, pi . , pi pi

pi pi pi.

3.3.3 ( )

pi pi R2 .

5... pi 1909 pi Syamadas

Mukhopadhyaya.

3.3. 61

pi , pi-

pi pi pi pi pi .

pi pi pi pi R2

` . pi pi, pi

pi . pi pi L pi

, pi pi Ps > 0 Ps < 0( Ps 0 Ps < 0 ). , s pi 0; `, pi pipi pi

pi p q . pi pi p q, s , ( 2.2.7),

. p q , Ps > 0 pi pi pi pi pi p

q, Ps < 0 . pi pi pi , r . pipi p; q r , pi

pi Ps > 0 Ps < 0. pi pi pi Ps > 0, pi Ps < 0 ( ). pi.

p q

a

3.5.

a L, a > 0 pi L a < 0 . pi Ps. a/ pi> 0 pi< 0, pi pi L pi. pi Z `

0

Ps. a/dt 0; (3.8)

pipi> 0 pi pipi < 0 .

pi Pns D st (. 2.2.1) piPs D .s/P s P D .s C ns/P;

pi (3.8) pi pi

.s C ns/ a, pi . `-pi

.t C `/ D .t/ t ;

pi pi t pipi pi t pi

`-pi:

t.t C `/ D P.t C `/ D P.t/ D t.t/:

62 3. pi

pi =2

ns.t C `/ D ns.t/; pi s.t C `/ D s.t/. .t C `/ D .t/ t , pi (3.8) Z `

0

P.t/dt D .`/ .0/ D 0:

pi, pipi .

3.3.1 3.1.2 .

3.3.2 1.1.7 (pi.

3.1.3).

3.3.3 pipi pi t D t0 ( 2.2.7) t D t0.

4pi

pi pi-

pi. pi pi , -

pi, pi

, pi pi pi pi -

pi. , pi,

pi pi pi

pi. pi pi pi

pi, pi pi pipi. pi

pi pi pi pi

( 5.6).

4.1 pi;

pi pi R3 pi R2

pipi , pi pi , pi,

pi , pipi pi-

pi pi pi pi .

pi , pipi pi

pi pi . pi

Rn n 1, n D 1; 2 3. , piU Rn , a U ,

pi u 2 Rn pi pi pi pi a pi U :

a 2 U ku ak < H) u 2 U:

63

64 4. pi

pi, Rn , pi

Dr .a/ D fu 2 Rn j ku ak < rg ; pi pi a r > 0. ( n D 1 pi , n D 2 .)

Dr.a/ D fu 2 Rn j ku ak rg , , pi

pi pi pi pi.. pi .a1 C r; a2; : : : ; an/ 2 Dr .a/ Dr.a/ (..

a1 C r C 2 ; a2; : : : ; an

).

pi, X Y pi Rn Rm, ,

pi f W X ! Y a 2 X X a pi f Y f .a/. , f

a , > 0, pi > 0

u 2 X ku ak H) kf .u/ f .a/k < : f X . -

pi .

,

: f , V Rn,

pi U Rm U \ X D fx 2 X j f .x/ 2 V g. f W X ! Y , 11 pi, pi

f 1 W Y ! X pi , f X Y .

pi pi pipi -

pi R3.

4.1.1

pi S R3 pi , p 2 S, pi pi U R2 pi W R3 pi pi p

S \W U . pi piS S \W , piW pi R3, pi S. W U ! S \W pi pi1 pi pi S \ W S. pi pi pi S

S.

4.1.2

pipi R3 pi pi pi pi

pi. , a pipi, p q

pi pi pipi . v

pipi, v a pi pipi, piv a D up C vq

1... pi surface patch pi

pi pi, pi .

4.1. pi; 65

pi u v. pi pi pi

.u; v/ D aC up C vq; pi

1.v/ D ..v a/ p; .v a/ q/ :pi 1 , -. ( pi pi).

pi pi pi pi-

pi pi.

4.1.3

R3 pi pi

pi (pi ) pi (pi

). pi, 1

z, pi ,

S D f.x; y; z/ 2 R3 j x2 C y2 D 1g: pi pi S :

.u; v/ D .cosu; sinu; v/: .u; v/ 2 S .u; v/ 2 R2, S . pipi, . 11 -

, .u; v/ D .uC2; v/ .u; v/. pi 11 pipi pi u 2 , 0 u < 2 . pi jV

V D f.u; v/ 2 R2 j 0 u < 2g 11, V pi R2 jV pi. pi R2 pi pi V

U D f.u; v/ 2 R2 j 0 < u < 2g; pi jU U pi., jU pi S, pi pi pi pipi pi

S x D 1, y D 0 ( 4.1). pi S,

pi. pi pi j QU , pi

QU D f.u; v/ 2 R2 j < u < g; pi pi pi S pi pipi pi S

x D 1, y D 0. S pi pi pi jU , j QU , f jU ; j QU g S S pi.

66 4. pi

4.1. pi x D 1, y D 0.

4.1.4

R3 pi pi pi (pi

) pi (pi ).

pi, 1 , pi

, :

S2 D .x; y; z/ 2 R3 j x2 C y2 C z2 D 1 : pi S2 pi -

pi ': p ,

pi pi p pi z pipi xy

, q. pipi , q p, '

q x. pi -

pi, pi

' .

pi pi pipi

p '. pi 0, p q

pi p sin .

p pi pi pipi xy 0, q

r , pi r pi pi q x. pi

kqk D cos , p kqk cos' D cos cos' kqk sin ' D cos sin ';

. pipi pi

p D .cos cos'; cos sin '; sin /:

4.1. pi; 67

x

y

z

qr

p

4.2. .

pipi , pi .; '/,

pi S2.

pi pipi , 11 ( pi)

.; '/ D .; 'C2/. pi pi

2

2; 0 ' 2:

.; '/ pi pi -

pi R2. pi pi-

pi

U Dn.; '/ j

2< 0 b c U . pi pi pi U pi pi b c, pi a. pi-

pi D pi S, pi p q, pi . (

, pi .)

pi . ( pi -

pi pi pi

.)

, , pi pi S[SC, pi pi pi pi pi W U ! R3,pi U D R2 n f.0; 0g, pi pi pi pipi xy:

.u; v/ D .u; v;pu2 C v2/:

pi pi pi , a pi S

pi pi. pi-

pi W U ! S \ W Q W QU ! S \ QW

70 4. pi

a

q

p

c

b

pi

4.6. pi .

pi a 2 S \ W \ QW . Q , 1.S \W \ QW / Q1.S \W \ QW / V U QV QU ,. 1 Q W QV ! V pi - pi Q . pi ,

Q . Qu; Qv/ D .. Qu; Qv// . Qu; Qv/ 2 QV .

u

U~

V~

~

V

U

4.7. pi .

4.1.1 pipi xy pi.

4.1.2 pi x W U ! R3 S2 x2 C y2 C z2 D 1 pi x y z:

x.u; v/ D .p1 u2 v2; u; v/:

pi U D .u; v/ 2 R2 j u2 C v2 < 1. pi

y

z ( U ) pi y

4.1. pi; 71

4.8. pi.

z, . pi pi

S2 pi ( 4.8).

4.1.3 pi

S D .x; y; z/ 2 R3 j x2 C y2 z2 D 1 : , ,

.x z/ cos D .1 y/ sin ; .x C z/ sin D .1C y/ cos pi S, pi pi

( 4.9). pi S pi pi

pi pi pi. ( pi-

pipi 4.1.3.)

S

, pi

. pipi

pi S pi .2

4.1.4 pi

pi, pi. ( pi -

pi pi.)

4.1.5 pi pi pi.

2... doubly ruled surface, pi pi pi pi-

. pi .

72 4. pi

4.9. pi pi.

4.2 pi

pi pi

pi ( ). pi, pipi

pi

pi . pipi pi

pi pipi .

, U pi Rm, pi-

f W U ! Rn pi n f , pi U ! R, pi . pipi pi f pi .

pi, m D 2 n D 3, f .u; v/ D .f1.u; v/; f2.u; v/; f3.u; v//;

@f

@uD@f1

@u;@f2

@u;@f3

@u

;

@f

@vD@f1

@v;@f2

@v;@f3

@v

pi . pi

:

@f

@uD fu; @f

@vD fv;

@2f

@u2D fuu;

@2f

@u@vD fuv;

@2f

@v@uD fvu;

@2f

@v2D fvv;

... pi pi pi fuv D fvu f .3

3... pi Schwarz.

4.2. pi 73

pi W U ! R3 . pi pipi .

4.2.1

pi W U ! R3 u v .u; v/ 2 U . , pipi u v pipi U .

pi pi pipi

4.4.

pi pi pi

.

4.2.2

S pi, pipi pi S

pi W U ! R3 pi Upi pi S. pi piS ,

p 2 S, pi pipi pi pi pipi p 2 .U /. A pipi pi pi S S A

pi S.

4.2.3

pipi 4.1.2 pi. ,

.u; v/ D aC up C vq pi u D p v D q , p q pi .

4.2.4

( 4.1.3) pi. ,

.u; v/ D .cosu; sin u; v/ pi

u D . sinu; cosu; 0/; v D .0; 0; 1/ pi .u; v/. jU j QU pi.

4.2.5

S2 4.1.4, pi

Q . D . sin cos'; sin sin '; cos /; ' D . cos sin '; cos cos'; 0/;

' D . cos2 cos'; cos2 sin '; sin cos /

74 4. pi

k 'k D j cos j. .; '/ 2 U , =2 < < =2, cos 0. pi pi pi Q .

4.1.2 pi pipi pi-

pi pi S2 ( pi

. 4.2.2). pi : pi pi

pipi pi ; pi -

pi pi pipi pi .

4.1.2 4.1.4

. pi pipi ( .

4.5.1) pi pi pi pi S

pi pi pi pipi pi .

pi pipi pi.

pi , pi

pi pi .

4.2.6

pi pi .

pi 5.6. pi pi

4.2.6.

4.2.7

U QU pi R2 W U ! R3 pi. W QU ! U 11 pi pi pi 1 W U ! QU , Q D pi.

pi Q pi pi-. , .u; v/ D . Qu; Qv/. ,

Q Qu D@u

@ Quu C@v

@ Quv; QQv D@u

@ Qv u C@v

@ Qv v;

pi

Q Qu QQv D@u

@ Qu@v

@ Qv @u

@ Qv@v

@ Quu v : (4.1)

pi pi pi

pi

J./ D@u@ Qu

@v@ Qu

@u@Qv

@v@Qv

. pi pi pi , Q pi R2,

J. Q / D J. Q/J./:( pi pi

pi Q pi pi Q .)

4.2. pi 75

D Q D 1 pi J.1/ D J./1. , J./ , , pi pi . 4.1

pipi Q . pi Q pi pipi pi-

, Q pi pipi. pipi pi-

Q , D Q 1. pi , W U ! S \ W Q W QU ! S \ QW

pipi pi pi S, V U QV QU pi pi .V / D Q . QV / D S \ W \ QW , D 1 Q W QV ! V 11 pi, , 4.2.6 . pi, Q pi pi .

pi pi pipi pi pi -

pi . , pi

pi pi

pi pi pi pi.

pi pi .

pipi pi

pi S, pi S

pipi pi S.

, pi pi pi-

pi , pi pipi S

pi pi pi S.4 pi

, pi pi

S pi, pi pi pi-

S. pi, pi S

5 (. 4.2.9). pi pi

, : pi 4.1.5

S pi pi , pi pipi pi.

4.2.1 , f .x; y/ ,

f.x; y; z/ 2 R3 j z D f .x; y/g pi pi pi pi

pi

.u; v/ D .u; v; f .u; v//:, pi pi . 5.6.4.

4... pi pi , . pi 2

. 3.5... pi .

76 4. pi

4.2.2 pi pi S2

4.1.2 . pi pi

pi .

4.2.3 pi pi pi ( pipi

u; v 2 R);(i) .u; v/ D .u; v; uv/,(ii) .u; v/ D .u; v2; v3/,(iii) .u; v/ D .uC u2; v; v2/.

4.2.4

x2

p2C y

2

q2C z

2

r2D 1;

pi p, q r , pi.

4.2.5 6 pipi pi C pipi

pi l pi C. pi pipi

xz, l z, a > 0 pi C pi l , b < a

C. pi pi

.; '/ D ..aC b cos / cos'; .aC b cos / sin '; b sin /:

4.10. .

4.2.6 pi pi pi

pi ( 4.11).

pi pi z, pi

.u; v/ D .v cosu; v sinu; u/;pi . pi pi -

pipi 7 p

z pi p pi z.

6... pi torus. pi.7... 4.5.

4.3. pi 77

4.11. pi.

4.2.7 .s/ pi R3 pi

pi. a > 0 pi pi

pi pi

.s; / D .s/C a.n.s/ cos C b.s/ sin /;pi n b .

pi pi. pi

pi pi pi a1. , , pi -

pi pi pi

pi 2a. pi :

pi .s; / pi

pi U R2, W U ! R3 11 ( ) . 5.6.3. pipi pi -

(.. 12.2.5).

4.2.8

R3 pi pi pi (. 4.12).

4.2.9 pi pi pi.

4.3 pi

pi f W S1 ! S2, pi S1 S2 pi. pi pi pi ,

pi pi pi -

.

pi pi pi pi ,

pi pi S1 S2 pi pi pi-

1 W U1 ! R3 2 W U2 ! R3, pi pi pi pi pi pi pi S1

78 4. pi

4.12. pi .

S2. 1 2 11 pi, pi pi f W S1 ! S2 pi-pi pi 12 f 1 W U1 ! U2, f pi ( pi -

R2 ). pi, pi Q1 W QU1 ! R3 Q2 W QU2 ! R3 pi 1 2, pi pi- 1 W QU1 ! U1 2 W QU2 ! U2, . pi pi Q12 f Q1 W QU1 ! QU2 12 f 1 W U1 ! U2 . ,

Q12 f Q1 D Q12 .2 12 / f .1 11 / Q1D . Q12 2/ .12 f 1/ .11 Q1/D 12 .12 f 1/ 1;

1 12 pi ( pi R

2). -

pipi pi pi pi

pi.

pi f W S1 ! S2 pi 11 pi pi pi f 1 W S2 ! S1 . pi ,8 S1 S2 -

pi . pi

.

4.3.1

f W S1 ! S2 . 1 pipi pi S1, f 1 pipi pi S2.

8... diffeomorphism. pi -

.

4.3. pi 79

pi pi pi S1 S2 pi pi pipi

pi 1 W U1 ! R3 2 W U2 ! R3, . pi f , f .1.u; v// D 2.F.u; v//, pi F W U1 ! U2 11 pi , F1 . pi pipi pi 4.2.6.

pi pi-

pi pi . pi f WS1 ! S2 pi pi , p 2 S1, pi piO S2 f .O/ pi S2 f jO W O ! f .O/ (pi pi pi pi . 4.2.9). pi

pi (piO D S1). pipi, 4.3.1 f pi , pi pipi pi

f 1 11.

4.3.2

pi pi pipi yz pi S pi

pi y pi z

pi pi pipi xy. pi pi

f .0; y; z/ D .cosy sin y; z/:

4.13. .

, f 11 pipi -

pi pi S. f pi -

pi pi pi pipi yz -

pi .u; v/ D .0; u; v/ pi f jU ; j QU g S 4.1.3. p D .0; a; b/ pipi yz. a pipi 2 , pi n

2n < a < 2.nC 1/ , f ..u; v// D .u 2n; v/ 2n < u < 2.nC 1/;

80 4. pi

pi pi f pi pi

O D f.0; y; z/ j 2n < y < 2.nC 1/g pipi pi

f .O/ D f.x; y; z/ 2 S j x 1g S. a pi pipi ,

j QU jU ( pi ). a pi pi pipi , pi ,

p pipi, pi pi O pipi pi

pi p pi f pi pi

S.

4.3.1 S pi, S ! R. , S pi, pi-

S ! R3 S ! R.

4.3.2 S x2 C y2 D z2 z > 0 (. 4.1.5). pi f pi pipi f.0; y; z/ j y > 0g S f .0; y; z/ D .y cos z; y sin z; y/. f pi .

4.4 pi pi

pi pi S () pi

S. pipi pi

pi.

4.4.1

pi pi S p 2 S pi p pi S pi pi p. pi TpS

S p pi S p.

pi TpS, pi pi-

W U ! R3 S p , .u0; v0/ D p. pi S pi p pi..t D t0, pi u.t/ v.t/

.t/ D .u.t/; v.t// (4.2) t t0, u.t0/ D u0, v.t0/ D v0. u v ( pi 5.6). ,

pi t 7! .u.t/; v.t// , . 4.2 pi pi S.

4.4. pi pi 81

4.4.2

W U ! R3 pi pi S pi pi p 2 S, .u; v/ U . pi S p - pi R3 pi pi pi u v (

pi pi .u0; v0/ 2 U pi .u0; v0/ D p).

pi pi S,

.t/ D .u.t/; v.t//: d=dt , pi

P D u PuC v Pv:pi P u v . ( - pi : u v u v pi u v

u.t/ v.t/ , pi

t ,

d=dt .)

, pi R3 pi pi

pi u v uCv pi .

.t/ D .u0 C t; v0 C t/ pi S, t D 0, p 2 S,

P D u C v : u v

pi p pi pi S.

pi , u v .

pi, pi pi

pipi. pi 4.4.1 pipi pi pipi

pi pi pi pi p, pi -

pi pi 4.4.2 (. 4.4.2). pi

u v pi pi pipi

pi .u0; v0/ pi pi pi-

pi pi, pi u 7! .u; v0/ v 7! .u0; v/( pi pi pi pi u D u0 v D v0).

pi pi pipi pi,

pi pi pi pi.

pi pi f W S ! QS pi. pi f p 2 S pipi pi pi f .p/ 2 QS p pi S, pi q. p

q pi , pi pi

pi S p. pi f p pipi

pi S p pi

82 4. pi

QS f .p/. , pi f p pipi piDpf W TpS ! Tf.p/ QS .

Dpf , w 2 TpS pi S p. , w pi

p pi S pi pi p, w D P.t0/. pi Q D f pi QS pi pi f .p/ t D t0, Qw D PQ.t0/ 2 Tf.p/ QS .

4.4.3

pipi, piDpf f p 2 S piDpf W TpS ! Tf.p/ QS piDpf .w/ D Qw pi w 2 TpS.

pi pi pi pipi

, Dpf .w/ pi f , p w:

pi pi (pi pi) pi pi w

p a priori Dpf .w/ pi pi pi pi.

W U ! R3 pi S pi pi p, p D .u0; v0/, ; U pi

f . .u; v// D Q ..u; v/; .u; v//:w D uCv pi p pi.t/ D .u.t/; v.t//,pi u v Pu.t0/ D Pv.t0/ D . pi QS Q.t/ D Q . Qu.t/; Qv.t//, pi Qu.t/ D .u.t/; v.t// Qv.t/ D .u.t/; v.t//,

Dpf .w/ D PQu Q Qu C PQvQv D . Puu C Pvv/ Q Qu C . Puu C Pvv/ QQv;pi pi u v pi t0. pi,

Dpf .w/ D .u C v/ Q Qu C .u C v/ QQv : (4.3) pi p, f , , pi p, f w, pi

pi.

pi . 4.3 pipi pi pi pi.

4.4.4

f W S ! QS pi pi p 2 S, piDpf W TpS ! Tf.p/ QS pi.

. 4.3 pi pi Dpf pi

fu; vg Tpf f Q Qu; QQvg Tf.p/ QS piu vu v

pi .u; v/ 7! ..u; v/; .u; v//.

4.4. pi pi 83

4.4.5

(i) S pi p 2 S, pi p pi- S ! S pi TpS ! TpS.

(ii) S1;S2 S3 pi f1 W S1 ! S2 f2 W S2 ! S3 pi, p 2 S1,

Dp.f2 f1/ D Df1.p/f2 Dpf1:

(iii) f W S1 ! S2 , p 2 S1 piDpf W TpS1 ! Tf.p/S2 .

pi (i) pi. (ii), w 2 TpS1 pi p pi 1 S1. pi 2 D f1 1 pi S2 pi Dpf1.w/ f1.p/, 3 D f2 2 D .f2 f1/ 1 pi S3 pi Df1.p/f2.Dpf1.w// f2.f1.p//.

pi 3 p pi Dp.f2 f1/.w/., (iii), g W S2 ! S1 pi f ,

g f f g pi S1 ! S1 S2 ! S2, .pi (i) (ii) pipi Df.p/g pi

Dpf .

pi pi pi pi pi pi-

pi .

4.4.6

S QS pi f W S ! QS pi, f - pi , p 2 S, piDpf W TpS ! Tf.p/ QS .

pi pi f pi p 2 S. pipi pi pi O S pi pi p -

f .O/ pi QS f jO W O ! f .O/ . 4.4.5(iii), Dpf (pi-

pi TpS D TpO). pi pi

pi 5.6.

4.4.1 pi pipi pi pi-

pi :

(i) .u; v/ D .u; v; u2 v2/, .1; 1; 0/.(ii) .r; / D .r cosh ; r sinh ; r2/, .1; 0; 1/.

84 4. pi

4.4.2 , .u; v/ pi, -

u v pi pi

.

4.4.3 S pi, p 2 S F W R3 ! R . rSF pi rF D .Fx ; Fy ; Fz/ F pi TpS. , pi S pi pi p

t D t0,

.rSF / P.t0/ D ddt

tDt0

F..t//:

pi rSF D 0 pi F S pi pi p.

4.4.4 f W S1 ! S2 pi pi S1. f pi S2.

4.5 pi

pi pipi TpS pi p 2 S pi R3, pi pipi pi

, pi

S p. , pi ,

4.4.2, pi pi W U ! R3 pi pi p, pi,

N D u vku vk(4.4)

(pi pi pi U pi p),

pi u

v . N pipi 9 -

pi p. pi pipi, N

pi pi pi pi p. , Q W QU ! R3 pi pi pi p, pi

4.2.7

Q Qu QQv D det.J.//u v;pi J./ pi pi pi Q .pi pipi Q

N Q DQ Qu QQv

k Q Qu QQvkD u vku vk

D N ;

pi pi pi J./.

pi .

9... standard unit normal.

4.5. pi 85

4.5.1

pi S pi pi A S

: pi pipi pi-

A, det.J.// > 0 pi .

pi pi pi pi:

4.5.2

S pi pi A pi

4.5.1, pi pi

S: pi pipi pipi -

pi A.

pi pi pi S pi

N , piN WS!R3( pi N

S ! R) , p 2 S, N .p/ - TpS. pi pi pi! pi-

pi S pi

.u; v/ uv pipi N S, pipi. pi pi pi

pi pi 4.5.1. pi

pi ( pi pi pi-

pi pi ). pi , pi pi-

pi S, pi pi S pi

pipi pi

S.

pi pi pi pi pi. -

pi pi .

4.5.3

Mobius pi pi pi

l pi p, p pi C

pi, p pi pi C, l

pi pi p. pi C x2 C y2 D 1 pipi xy, l 1 pi pi

z p D .1; 0; 0/, p pi pi z, l pipi pi =2 pi

p pipi pi pi p z. l pi

.1; 0; t/

.t; / D

1 t sin 2

cos ;

1 t sin

2

sin ; t cos

2

:

pi

U D f.t; / 2 R2 j 1=2 < t < 1=2; 0 < < 2g:pi pi Q pi pi

86 4. pi

pi pi QU D f.t; / 2 R2 j 1=2 < t < 1=2; < < g: pi pi pi, pi - Mobius pi pi pi

pi. Mobius pi-

S (. 4.5.1).

4.14. Mobius.

pi pipi N

(pi t D 0).

t D sin

2cos ; sin

2sin ; cos

2

; D . sin ; cos ; 0/;

pi

t D cos cos

2; sin cos

2; sin

2

:

, N .

Mobius pi pi -

N S pi pi

pi S. pipi , .0; / ,

N D ./N ;pi W .0; 2/! R ./ D 1 . pi ./ D C1 2 .0; 2/, ./ D 1 2 .0; 2/. N N pi, pi pi D 1. .0; 0/ D .0; 2/ ( N )

N D lim#0

N D .1; 0; 0/ pi N D lim"2

N D .1; 0; 0/:

pi, Mobius pi.

pi S pi, pi-

pi S. pi

pipi .

p 2 S N pi p. pi pi pi pipi TpS

4.5. pi 87

, pi -

pi TpS,

N . , pi N pipi pi

TpS: pi pi pi N . pi,

TpS pi pi, pi

pi TpS .

TpS

N

4.15. pi.

v w TpS, pi

( pi pi pi ) v w

pi pipi pi v , pi

pi pipi w.

cvw. cwv D cvw;

pi cvw pi - TpS. pi cvw pipi pipi 2 .

4.5.4

p 2 S2 pi pipi p. p ,

p p, , p, -. pi pi S2: pi

pi .

pi, pi pi , pi

p .1; 0; 0/ , pi

pi v D .0; 1; 0/ w D .0; 0; 1/ cvw D =2( 3=2; 7=2 pi.). pi ,

pi pi.

88 4. pi

4.5.1 pi pi pi-

Mobius 4.5.3. pi

R2,

pi C1 1 .

4.5.2 pi pi S QS S pi. pi QS pi.

3pi pi

1.1.6 : .t/ D .p cos t; q sin t/. (i) - pi p.1 cos t/. (ii) n D .q cos t; p sin t/ pi .t/ pi

j.p f1/ nj=knk j.p f2/ nj=knk.1.2.3 pi 3.2.2.

1.4.5 (ii) (i) pi T1 0 T1-pi. fTr T1g. (iii) pi .

1.4.6 .t/ D .1.t/; : : : ; n.t//. i T -pi, pi, 1.4.5, i , () pi, Ti .

1.4.4, i , T D kiTi pi ki . k pi pi ki .

pi T0 D T=k.1.5.2 1.5.1 f

.@f@x;@f@y;@f@z/ pi f .x; y; z/ D 0 pi

pi pipi . pi

5.6.

1.5.3 .

2.2.3 s pi pi pi .

2.2.10 pi .s/ pi

. pi s, pi .s/

p C sa `, pi .s/ `. pi .s/. P.s// D a. :(i) S n n pi ( S

S t S) v n 1 pi, vtSv D 0.

373

374 3. pi pi

(ii) A pi ( AtA D I ) - pi s, At dA

ds ( dA=ds

pi A).

2.3.2 pi pi .

2.3.3 pi a D t cos C b sin .2.3.4 a r , .a/.a/ D r2

pi pi. C nC Pb.2.3.6 pi pi pi pi -

vi vj pi .

3.2.1 pi pi 1.

3.3.1 pi 2x1x2 x21 C x22 .4.1.4 pi pi U .

5.2.3 pi pi : .t/ D aC tb. pi- . 5.1 pipi t .

pi pi

pi pi .

5.6.1 pi 4.2.6.

5.6.4 .u; v/ D .f .u; v/; g.u; v/; h.u; v//, pifu fvgu gv

-

. pi 4.2.6 pi V

W F1 W V ! W F1.f .u; v/; g.u; v// D .u; v/ .u; v/ 2 W . '.x; y/ D h.F1.x; y//.

6.1.5 pi pi pi u D u0 R v1v0

pG.u0; v/dv.

6.3.8 pi .2.5.

6.4.3 pi pi

.

6.5.3 pi pipi pi p pi.

6.5.4 Mobius pi (i) b 2 R.pi Mobius -

Mobius.

7.1.2 pi .u N /u, pi Nu Nv u v , pi N

.

7.3.4 pi 6.4.7.

7.4.3 pi 7.4.7.

8.1.6 2 2 pi

A Da c

b d

pi A2 .aC d/AC .ad bc/I D 0.

8.1.8 pi 8.1.6.

8.2.3 pipi pi EN D GL EN GL.

3. pi pi 375

8.2.5 pi 8.2.2.

8.2.6 (i) . 8.5.

8.2.7 pi 8.1.2, pi 8.2.1

8.1.1.

8.2.8 pi 6.1.4 7.1.3.

8.3.1 (iii) pi pi .

8.4.1 pi 8.2.4 7.3.3 8.4.3.

8.5.2 pi 8.1.1.

8.5.3 pi 8.2.4 pi 8.1.2.

9.1.1 pi 9.1.4 9.1.6.

9.1.3 pi 4.2.7.

9.1.4

x2

p2C y

2

q2C z

2

r2D 1

pi .t/ D .f .t/; g.t/; h.t//

. Rf ; Rg; Rh/ D f

p2;g

q2;h

r2

pi .t/.

9.1.5 pi 8.2.2.

9.2.2 pi 6.2.1.

9.3.3 pi , pi w > 1,

v pi pi . 9.14 pipi 2 .

9.4.2 S2 pipi pi pi p q.

9.4.3 (i) pi LHospital.

9.5.2 (ii) , pipi p

R, q0. pi pi p q0.

10.1.1 pi pi piWeingarten.

10.1.3 P D Lw2.10.2.2 pi 6.1.4 9.5.1.

10.2.4 pi , 9.5.1 10.2.3(ii).

10.3.1 pi pi 10.3.4, 1 pi

p pi, 2 D 2H 1 pi .

10.4.2 pi

2 0

0 1

.

10.4.3 pi 9.2.1.

11.1.4 pi pi 11.1.4 .2.3. pi-

2R pi pi i.b C r/ i.b r/, pi.11.2.1 pipi pi a b .

11.2.2 pi 11.1.1.

376 3. pi pi

11.2.3 pi pipi pi l -

pi 11.1.3.

11.2.4 (i) P H, pi pi pi

P l m. (iii) l m pi 11.2.3,

F H. pi pi 11.2.3 pipi

pi G pi pi F.i/ i

F.l/ l . pi pi (ii).

11.2.5 pi .2.2.

11.3.2 pi 11.2.4.

11.3.3 pi 11.2.5.

11.3.4 pi 11.2.5.

11.3.6 pi pi

pi. pi pipi

pi pi a pi pi

b c pipi pi

b c, pipi pi .

11.3.8 pi pipi pi -

11.3.7 pi pi 11.3.6.

11.4.2 H pi l .

11.4.3 DP pi a D 0, b 2 R.11.5.3 pi pi M.z/ D .a; bI c; z/ Mobius

pi pi .a; b; c/ .1; 0; 1/. pi M Mobius pi pi .a; b; c/ .1; 0; 1/, M.z/ D .1; 0I 1;M.z// D .a; bI c; z/.

12.1.1 pi 8.2.9.

12.1.3 pi 8.6.1 12.1.1.

12.2.4 pi 8.5.1.

12.3.1 pi 8.1.6 8.2.9.

12.5.1 pi D e Q .12.5.3 (ii) pi pi Weierstrass . 12.25 12.26.

13.1.2 pi 13.1.2.

13.3.2 k.t/ D .nt k/, pi pi 8.5.3(iii) 'k D k=. 1 C C n1/.

13.5.2 E 12V.V 1/.13.7.1 13.7.2 pi

de Moivre.

13.7.3 Q pi Q , d=ds D .1 2/1=2 P, pi D cos . pi pi Green

R `./0 .d=ds/ds D 0.

13.8.1 pi ( pi )

vtHv < 0 ( > 0) 2 1 pi v.

4

1

1.1.1 pi pi x 0.1.1.2 (i) .t/ D .sec t; tan t/ =2 < t < =2 =2 < t < 3=2.

pi :

pi pi pi , y 1 y 1.(ii) .t/ D .2 cos t; 3 sin t/.

1.1.3 (i) x C y D 1. (ii) y D .ln x/2.1.1.4 (i) P.t/ D sin 2t.1; 1/. (ii) P.t/ D .et ; 2t/.

4.1.

377

378 4.

1.1.5 P.t/ D 3 sin t cos t. cos t; sin t/ sin t D 0 cos t D 0, t D n=2 pi n . ( 4.1) (. 1.3.3.)

1.1.6 (i) pi p pi .p cos t p/2 C q2 sin2 tD .p2q2/ cos2 t2p2 cos tCp2 D p2.1 cos t/2, pi 2p.

(ii) P D .p sin t; q cos t/, n D .q cos t; p sin t/ n P D 0. pi pi pi .t/

.p cos t p; q sin t/ nknk D

pq.1 cos t/.p2 sin2 t C q2 cos2 t/1=2

p2q2.1 2 cos2 t/.p2 sin2 t C q2 cos2 t/ D q

2:

(iii)

.p f1/ nkp f1k

D .p f2/ nkp f2k:

pi pipi q.

1.1.7 pi t , .at; a/, pi

pi .0; 0/ pi

.a.t sin t/; a.1 cos t// ( 4.2).

ta

at0

4.2.

1.1.8 pi .x; y; z/ x2 C y2 D 1=4 -

x C 12

2C y2 C z2 D 1. pi 1 z 1, pi

z D sin t . pi xC 14 C sin2 t D 34 , x D12sin2 t D cos2 t 12 . pi pipi pi pi y D sin t cos t (y D sin t cos t , pipi pi t 7! t).

1.1.9 P D .2 sin t C 2 sin 2t; 2 cos t 2 cos 2t/ D p2.p2 1; 1/ t D =4. pi y . 1p

2 1/ D .x p2/=.p2 1/

y . 1p2 1/ D .x p2/.p2 1/.

1.2.1 P.t/ D .1; sinh t/, k Pk D cosh t s D R t0 cosh udu D sinh t .

4. 379

1.2.2 (i) k Pk2 D 14 .1C t/C 14 .1 t/C 12 D 1. (ii) k Pk2 D 1625 sin2 t C cos2 t C 925 sin2 t Dcos2 t C sin2 t D 1.

1.2.3 d=d , P D . Pr cos r sin ; Pr sin C r cos /, k Pk2 D Pr2 C r2. pi r D Pr D 0 pi . Pr2 D 1 r2, pi pi pipi r D sin.C/ pi . pi 1=2, 3.3.

1.2.4 u , j P uj D k Pk cos , pi P u, P u k Pk. pi

.q p/ u D ..b/ .a// u DZ ba

P udt Z bak Pkdt:

u D .q p/=kq pk pi.1.3.1 (i) P D sin 2t.1; 1/ t pipi =2,

. (ii) , P 0 0 < t < =2. (iii) P D .1; sinh t/ pi pi , .

1.3.2 x D r cos D sin2 y D r sin D sin2 tan , pi - 7! .sin2 ; sin2 tan /. 7! sin 11 pi pi pi .=2; =2/ .1; 1/, t 7! sin1 t , t D sin pi pi. sin2 D t2, sin2 tan D t3=

p1 t2, pi pi pi

.

1.3.3 (i) P D 0 t D 0() m n 2. m > 3, pi R 0 t D 0, R R t D 0 pi n > 3. pi pi pipi: .m; n/ D .2; 2/ .3; 3/ R t D 0, pipi pi pi .m; n/ D .2; 3/ .3; 2/ pipi pi R t D 0. (ii)pi pi .t/ D

t2; t

3p1t2

pi P D 0,

R D .2; 0/ D .0; 6/ t D 0, . (iii) Q.Qt/ pi .t/ t D t0, t D t0

d Q=d Qt D .d=dt/.dt=d Qt / D 0;d2 Q=d Qt2 D .d2=dt2/.dt=d Qt /2 d3 Q=d Qt3 D .d3=dt3/.dt=d Qt /3 C 3.d2=dt2/.dt=d Qt /.d2t=d Qt2/:

pi dt=d Qt 0, pi d2 Q=d Qt2 d3 Q=d Qt3 t D t0.

1.3.4 (i) Q.t/ D .'.t//, pi pi '. pi Q. .t// D .'. .t/// D .t/. (ii) Q.t/ D .'.t// O.t/ D Q. .t//, pi' pi pi, O.t/ D ..' /.t//. ' pi pi d

dt.'. .t// D P'. .t// P .t/ 0,

P' P 0.1.4.1 .t C 2/ D .t/ t . pi .t/ D .u/,

cos3 t.cos 3t; sin 3t/D cos3 u.cos 3u; sin 3u/. cos3 t D cos3 u, cos t D cosu, u D t; t; C t 2 t ( pi pipipi 2). pipi

380 4.

t D n=3 pi n, t . pi pi , t D =3; 2=3, pi pi pi u D 2=3; =3, . pi

.=3/ D .1=8; 0/.

1.4.2 pi Q.t/ D .cos.t3 C t/; sin.t3 C t// pi

.t/ D .cos t; sin t/ .

1.4.3 T -pi, kT -pi k 0 ( pi pi k k > 0, k k < 0). T1-pi T2-pi, k1T1-pi k2T2-pi

k1; k2, .tCk1T1Ck2T2/ D .tCk1T1/ k2T2-pi, pi .t/ k1T1-pi.

1.4.4 T -pi. T D kT0 C T1, pi k 0 T1 < T0. 1.4.3 T1-pi T1 > 0 pi T0.

1.4.5 (i) pi T1 > 0 T1-pi, T1 -

, pi T2 > 0 -

T2-pi. pi -

. (ii) fTr gr1 (i) , pipi pi T1 0. pi T1-pi ( ) .t C T1/ D limr!1 .t C Tr / Dlimr!1 .t/ D .t/. 1.4.3, .Tr T1/-pi r 1, 0. (iii) fTr g (i) Tr ! 0 r !1, pi pi0 D .f .t C Tr / f .t//=Tr D Pf .t C Tr / pi 0 < < 1. r !1pi Pf .t/ D 0 t , f .

1.4.6 pi, T0 D .ki=k/Ti pipi Ti , i T0-pi. T pipi pi

fTi ; 2Ti ; : : : ; kiTi g i pi i , P D fT 0 2 T j T 0-pig. P pipi ( T pipi-) ( T 2 P). P T 00 pi T

00-pi ( T

0-pi, T 0 > T T 0 2 P). 1.4.4, T0 D k0T 00 pi k0, pi pi k0i T

00 D k0iTi i pi

i . pi kiTi=k D k0kiTi , kk0 ki . k , k0 D 1, T0 D T 00.

1.5.1 x.1 x2/ 0() x 1 0 x 1, pi pi pi () . pi t 1 0 t 1, pi pi y 0.

1.5.2 .t/ D .x.t/; y.t/; z.t/ pi pi f .x; y; z/ D 0, pi- f .x.t/; y.t/; z.t// D 0 pi t pi Pxfx C Pyfy C Pzfz D 0, P rf .fx; fy ; fz/. pi pi,

rf pi. pi f D 0 g D 0 pi rf rg pi .

1.5.3 .t/ D .u.t/; v.t/; w.t// pi R3. pi Pu, Pv Pw t . pi Pu.t0/ 0 x0 D u.t0/. pi pi 1.5.2, pi h.x/ x x0 t D h.x/ x D u.t/ t t0. pi , t t0, .t/ pi

4. 381

pi f .x; y; z/ D g.x; y; z/ D 0, pi f .x; y; z/ D y v.h.x// g.x; y; z/ D zw.h.x//. f g pi pi- , rf D .Pvh0; 1; 0/ rg D . Pwh0; 0; 1/, pi d=dx.

2

2.1.1 (i) ( 1.2.2(i)), D k Rk D k.14 .1 C t/1=2;14 .1 t/1=2; 0/k D 1p8.1t2/ . (ii) ( 1.2.2(ii)),

D k Rk D k.45 cos t; sin t; 35 cos t/k D 1. (iii)pi 2.1.2 D k.1;sinh t;0/.0;cosh t;0/kk.1;sinh t;0/k3 D

cosh t

cosh3 tD sech2t .

(iv) .3 cos2 t sin t; 3 sin2 t cos t; 0/.3 cos2 tC6 cos t sin2 t; 6 sin t cos2 t3 sin3 t; 0/ D.0; 0;9 sin2 t cos2 t/, D k.0;0;9 sin2 t cos2 t/kk.3 cos2 t sin t;3 sin2 t cos t;0/k3 D

13j sin t cos t j . pi-

t pipi =2, .1; 0/ .0;1/ .

2.1.2 pi 1.3.5, v.t/ ()

t , kv.t/k () t pi pipi v.t/ t . pi pi pi

pi 2.1.2. pi pi .t/ D .t; t3/ .t/ D 6jt j=.1C 9t4/3=2, pi pi t D 0.

2.2.1 t ns D 0 pi Pt D sns .2.2.2 , t D P Pt ns (

ns pipi pi t). pi s D Pt ns .2.2.3 pi , pi 2,

Qs D s M pi l . : pi '0 2.2.1 l

x, ( pi ) Q' D '. , Q pi, pi pi

pi pi. pi pipi,

2.3.5, pi ,

pi pi pipi pi pi

pi, pi . pi -

pi .

2.2.4 pi pi s pi-

pi a ' pi .

s s C s. - ( pi ) .T cos '/, pi

pipi , T cos ' pipi , . -

.T sin '/, pipi

, pi pipi s. pi T sin ' D s C pi ; , pipi ' D s D 0 - C. pi T cos ' D T sin ' D s pi tan' D s=a,pi a D =. pi sec2' d'

dsD 1=a, pi pi

382 4.

s D d'=ds D 1=asec2' D 1=a.1C tan2 '/ D 1a .1C s2=a2/1. pi pipi pi 2.2.4.

2.2.5 d=dt D d=dt C dns=dt D .1 s/ds=dt t, s

pi ds=dt D j1 sjds=dt . pi

t D .d=dt/.ds=dt/ D t, pi n D ns . pi, pi pi s pi s n

s D d t=ds D .d t=dt/=j1 sj.ds=dt/ D j1 sj1 d t=ds

D s.1 s/1ns D s.1 s/1ns D sj1 sj1ns :2.2.6 .s0/ pi .s0/

.s0 C s/. pi

..s0/1

2..s0/C .s0 C s/// ..s0 C s/ .s0// D 0:

pi Taylor pipi pi pi

s , pi (pi pi-

s0) . / Ps C 12 . R 1 R/.s/2 D 0. pipi pi s s pi pi . / P D 0 . / R D 1. pi pi pipi D C ns pi , , pi R D sns , pi pipi D 1=s .

2.2.7 pi tC 1s .st/Ps2sns D Ps

2sns , pi

u D R kPkds D R Ps2sds D u0 1s , pi u0 . pi

pi ns pi t. dns=du D st=.du=ds/ D

3s

Ps t, pi-

pi 3s = Ps .pipi .s/ .s/C ns.s/ pi . pi .s/, pi D 1=s.s/, pi pi pi pi

ns.s/, pi .s/.

d=dt , 0 D a.1cos t; sin t/, pi s pi ds=dt D 2a sin.t=2/, t D d=ds D .sin.t=2/; cos.t=2//.pi ns D . cos.t=2/; sin.t=2// Pt D .d t=dt/=.ds=dt/ D 14a sin.t=2/ .cos.t=2/; sin.t=2// D 1=4a sin.t=2/ns , pi pi 1=4a sin.t=2/

.t/ D a.t sin t; 1 cos t/ 4a sin.t=2/. cos.t=2/; sin.t=2//D a.t C sin t;1C cos t/:

pi Qt D Ct , pi a.Qtsin Qt ; 1cos Qt/C a.;2/, pipi pi pi

a.;2/.2.2.8 pi .s/ ` s,

pi .s/. pi P PC.` s/ R D s.` s/ns ( d=ds). v pi dv=ds D s.` s/, pi pi ns pi t. dns=dv D 1s.`s/ Pns D

1`s t, pi pi 1=.` s/.

4. 383

2.2.9 pi Q.s/ D .sinh1 s;p1C s2/. -

.s/ D Q.s/s PQ.s/ Dsinh1 s sp

1Cs2 ;1p1Cs2

D .utanh u; sechu/

u D sinh1 s. pi, .x; y/ , u Dcosh1.1=y/ x D cosh1.1=y/

p1 y2.

2.2.10 pi .s/ pi pi .s/ ` pi q .s/ pi pi .s/

pCsa. pi .s/.pCsa/ D .s/.q.s//, pi pi pipi . P .s/ D aC

dds.s/

.q.s//.s/ P.s/.

pi pi a , -

. pi pi . dds.s//.q.s//.s/.q.s// D 0.

A D .s/, v D q .s/, pipi ( pi ) .Av/t dA

dsv D 0, vtAt dA

dsv D 0. A , pipi pi

(i) (ii) pi. pi (i) pi :

vtSv D Pi;j vivjSij D Pi;j vj viSj i D Pi;j vivjSij . (ii) pi AtA D I .

2.3.1 (i) t D .12 .1 C t/1=2;12 .1 t/1=2; 1p2 // ,

Pt D .14 .1Ct/1=2; 14 .1t/1=2; 0/, D kPtk D 1=p8.1 t2/

n D 1 Pt D 1p2 ..1 t/1=2; .1C t/1=2; 0/ b D t n D .12 .1C t/1=2; 12 .1 t/1=2; 1p2 //

Pb D .14 .1C t/1=2;14 .1 t/1=2; 0//, D 1=p8.1 t2/.

Pn D t C n pi .(ii) t D .45 sin t; cos t; 35 sin t/ , Pt D .45 cos t; sin t; 35 cos t/, D kPtk D 1 n D 1 Pt D .45 cos t; sin t;35 cos t/ b D t n D .35 ; 0; 45 //, Pb D 0 D 0. pi 2.3.5, 1= D 1 C 1 n D .0; 1; 0/ pipi pi pi .0; 1; 0/ b D .35 ; 0;45 /, pipi 3x C 4z D 0.

2.3.2 a D =.2 C 2/, b D =.2 C 2/. 2.1.3 2.3.2, pi a b pi a=.a2 C b2/ D b=.a2 C b2/ D . 2.3.6, pi pi pipi .

2.3.3 t a (= ) pi n a D 0 t, n b pi- R3, a D t cos Cb pi a , D sin pi a D t cos b sin pi D cot . , D , pi D cot pipipi a D t cos C b sin , pi t a D cos , t a. 2.1.3,

pi d=d D .a sin ; a cos ; b/ z cos1.b=

pa2 C b2/.

2.3.4 pi .a/ .a/ D r2 pi t .a/ D 0t tC n . a/ D 0, pi n . a/ D 1= n tC .t C b/ . a/ D P=2,pi b . a/ D P=2 b t n . a/ D . P=2/P = D . P=2/P. , . (2.22), D . P/P, pi

384 4.

.2 C . P/2/P D 2 P C 2. P/. P/P D 0, 2 C . P/2 , r2 (pi r > 0). a D C n C Pb . (2.20) pipi a D t C Pn C .t C b/ C . P/PC . P/.n/ D 0, a - k ak2 D 2 C . P/2 D r2. pi pi a r .

2.3.5 P D P P, T D P t k P k2 D .P P/ .P P/ D P P P -. pi R D P R pi pi pi , pi N D Pn. pi B D P t Pn. P ( pi), P.t n/ D Pb, P , P t Pn D Pb ( .1.6).

2.3.6 ij D vi vj . v1, v2 v3 ij D ij (D 1 i D j D 0 i j ).1 pi ij D ij s, s D s0. vi vjpi ij D

P3kD1.aikkj C ajkik/. ij D ij

aij Caj i D 0. , pi

s D s0.

3

3.1.1 P D . sin t a sin 2t; cos t C a cos 2t/, k Pk2 D 1 C a2 C 2a cos t . 1 C a2 2jaj D .1 jaj/2, jaj 1. jaj D 1 k Pk D 2.1C a cos t/, . a D 0

. 0 < jaj < 1, .t1/ D .t2/ H) 1 C a cos t1 D 1 C a cos t2H) cos t1 D cos t2 H) t2 D t1 2 t1. pipi .t2/ D..1Ca cos t1/ cos t1;.1Ca cos t1/ sin t1/, .t1/ D .t2/ H) sin t1 D 0H) t1 D 0 . pipi t2t1 pipi 2 , pi pi 2 . jaj > 1, pi cos t D 1=a, pi 0 t < 2 , t1 < t2, . pi pi 1.1.7 (pi

pipi a D 2), pi - pi pi pi pi t1 t t2, .

3.2.1 1, M R2 M.v/ D P vCb,pi P 3 3 pi b . Q DM./, PQ D P P, k PQk D k Pk, pi Q . - pi pipi xy R3, . 3.2 pi

: A./ D R T0 . P R/ k dt , pi k D .0; 0; 1/. pi .1.6 pi

Q pi (pi M , pi, pi-

pi Q pi pi).1... ij Kronecker, pi :

ij D1 i D j0 i j :

4. 385

3.2.2 pi pi .t/ D .p cos t; q sin t/, 0 t 2 ,

R 20

qpq sin2 t C pq cos2 tdt D 2ppq.

pi , ` pi

` p4 pq D 2ppq, , p D q. R 20 k Pkdt DR 20

qp2 sin2 t C q2 cos2 tdt:

3.3.1 .x1; y1/ .x2; y2/ , xip2

C yiq2

< 1 i D 1; 2. pi .tx1 C .1 t/x2; ty1 C .1 t/y2/ pi 0 t 1.

.tx1 C .1 t/x2/2p2

C .ty1 C .1 t/y2/2

q2

D t2 x21p2

C y21

q2

!C .1 t/2

x22p2

C y22

q2

!C 2.1 t/

x1x2

p2C y1y2

q2

< t2 C .1 t/2 C t.1 t/ x21p2

C y21

q2C x

22

p2C y

22

q2

!

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CH1-4, Διαφορικη΄Γεωμετρία