Διαφορική Γεωμετρία Καμπυλών Και Επιφανειών

. .

2009

:

http://www.math.uoa.gr/evassilhttp://www.math.uoa.gr/mpapatr

.

COPYRIGHT 2009by E. Vassiliou M. Papatriantafillou

[email protected], [email protected] rights reserved

,

, -

.

D. Hilbert 2o -

, 1900 (.

C. Reid [16, . 170])

-

-

,

.

-

()

, , , -

. ,

. , -

, -

.

,

. .

.

v

vi

.

, -

.

,

. , ,

.

() ,

, .

-

, , -

,

. [2],

[16], [17] .

, , .

,

.

.. .. , 2009.

v

1 1

1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 . . . . . . . . . . . . . . . . . . . . . 2

1.2 . . . . . . . . . . . . . . . . . . . 7

1.3 . . . . . . . . . . . . . . . . . . . . 9

1.4 . . . . 20

1.5 Frenet . . . . . . . . . . . 22

1.6 . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.7 . . . . . . . . . . . . . . . . . . . 27

1.8 . . . . . . . . . . . . . . . . . . . . . . . 31

1.9 . . . . . . . . . . . . 34

1.10 . . . . . . . . . . . . . . . . . . . . . . . 40

1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2 81

2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.4 . . . . . . . . . . . . . . . . . . . 102

2.5 . . . . . . . . . . 106

2.6 . . . . . . . . . . . . . . . . . . . . . . 113

2.7 . . . . . . . . . . . . 116

2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3 Gauss 141

3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.1 . . . . . . . . . . . . . . . . . . . . . . . . 142

3.2 . . . . . . . . . . . . . . . . . . . 145

3.3 Gauss . . . . . . . . . . . . . . . . . . . . . . . 150

vii

viii

3.4 . . . . . . . . . . . . . . . . . . 154

3.5 . . . . . . . . . 157

3.6 . . . . . . . . . . . . 159

3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

189

199

201

1

1.0

, , -

.

, , , , ..

( )

,

.

,

. -

() .

. -

.

R. Descartes, P. Fermat C. Huygens,

3.

-

, -

.

, . -

1

2 1.

I. Newton, G. W. Leibniz, L. Euler, G. Monge, J. Bernoulli, A. C. Clairaut,

F. Frenet, J. A. Serret, J. Bertrand, Ch. Dupin, .

.

,

, , , -

, ..

-

"" .

1.1

X - ( ) . -

,

(, , .),

f : R R, , ... X, . ,

:

(parametrized curve),

(space curve) : I R3, I R .

"" ( )

(-

) t I., t I, (t) R3,

(1.1.1) = (1, 2, 3).

( )

i : = ui ; i = 1, 2, 3, ui : R

3 R i-, .

i.

(I) R3, (plane curve). , (I) R2.

( ), , X = (I). (I).

1.1. 3

: R R2 : t 7 (2

2t,

2

2t),

: R R2 : t 7 (t, t) : R R2 : t 7 (t3, t3)

: R R2 : t 7{(t, t) : t 0(t2, t2) : t > 0,

(R) = (R) = (R) = (R) = R, R2 (. ).

x

D

y

1.1

,

, (t) = 1, t R, , (t) 6= 0, t R, , 0, (0) = (0, 0),

.

: (1) R ()

(2) ( )

R. ,

,

.

4 1.

: I R3 to (to) 6= 0. (tangent line) (to),

(1.1.2) (s) = (to) + s(to) , s R.

x

y

( )ota

( )ota

1.2

(to) to - .

, -

.

(regular), , ( ).

-

( 3), . ,

, 11. ,

, " "

, Cr-, r 3.

,

(1.1.3) (t) = (1(t), 2(t),

3(t))

1.1. 5

(tangent vector)

(velocity) (t),

v(t) := (t) = (1(t)2 + 2(t)2 + 3(t)2)1/2 (speed) t.

,

(1.1.4) (t) = (1(t), 2(t),

3(t))

(acceleration) t.

: I R3, (length)

(1.1.5) L() :=

I(t)dt.

, : I R3, < , > -

(1.1.6) < , > : I R : t 7< (t), (t) > (1.1.7) : I R3 : t 7 (t) (t). -

< , > (t) =< (t), (t) > + < (t), (t) >,(1.1.8)

( )(t) = ((t) (t)) + ((t) (t)).(1.1.9) :

[. (1.1.1)]

,

< (t), (t) >= 1(t)1(t) + 2(t)2(t) + 3(t)3(t),

(t) (t) =(2(t)3(t) 3(t)2(t), 3(t)1(t) 1(t)3(t), 1(t)2(t) 2(t)1(t)

),

.

(I),

6 1.

1.1.1 . (1) : I R3 . (t) = 0, t I, .

,

(t) = 0 (t) = (t) = t+ . .

(2) 0. (to) 0 (to) 6= 0, (to) (to).

, t I (t) 0,

: I R : t 7 (t)2 = 1(t)2 + 2(t)2 + 3(t)2 =< (t), (t) > . to,

(to) = 0, , (1.1.8),

0 = (to) =< (to), (to) > + <

(to), (to) >= 2 < (to), (to) >,

(to) (to).(3) : I R3 , (t) 6= 0,

t I. (I) 0, (t) (t), t I.

, (t) 0

: I R : t 7< (t), (t) > . , = 0.

(t) =< , > (t) =< (t), (t) > + < (t), (t) >= 2 < (t), (t) > .

= 0 (t) (t), t I, .

(4) P Q P , Q.

p q P Q , P , Q

: [0, 1] R3 : t 7 (t) := p+ t(q p) [. (1.1.5)]

L() =

10(t)dt =

10q pdt = q p.

1.2. 7

: [x, y] R3 (x) = p (y) = q. u R3 u = 1,

< a, u > (t) = < (t), u > + < (t), u > = < (t), u >

(t)u = (t),

L() =

yx(t)dt

yx< (t), u > dt = < (t), u > |y

x

= < q, u > < p, u > = < q p, u >,

u R3 u = 1. u = (q p)/q p, -

L() / q p = q p = L().

1.2

: I R3 , : J R3 (reparametrization) , h : J I, = h (. ).

I - X R3

J

h

6

-

1.1

" h " h 1 1 .

, h 1 1 h(s) 6= 0, s J .

. , .

X := (I) = (J) X.

8 1.

1.2.1 . , L() = L().

. : [a, b] R3 : [a, b] R3 = h, h : [a, b] [a, b] .

L() =

ba(s)ds =

ba( h)(s)ds

=

ba(h(s)) |h(s)|ds.

h > 0, h ,

L() =

ba(h(s))h(s)ds =

h(b)h(a)

(h(s))dh(s)

=

ba(t)dt = L().

h < 0, h ,

L() = ba(h(s))h(s)ds =

h(b)h(a)

(h(s))dh(s)

= ab(t)dt = L(),

.

. , . ,

, X = (I)., ,

, 1.

1.2.2 . : I R3 - : J R3 (s) = 1, s J .

. I := [a, b].

s : [a, b] [0, L()] : t 7 ta(u)du.

1.3. 9

,

s(t) = (t) = v(t) > 0, (. ),

h : [0, L()] [a, b]

(1.2.1) h(s) = 1/s(h(s)), s [0, L()]. := h,

(s) = (h(s)) |h(s)| = (h(s))/|s(h(s))| = 1, .

: J R3 (s) = 1, s J , . -, , ()

.

.

1.3

X = (I), . , X . , - ,

.

(s), s J , .

(1.3.1) T (s) := (s) s J.

, T (s) (s). T (s) = 1, s J , T (s) R3.

s J

(1.3.1) T : J s 7 T (s) = (s) R3.

10 1.

1.3.1 . (s), s J , ,

T T,

T (s) T (s), s J.. T (s) , - T (s)2 =< T (s), T (s) > ,

< T, T > (s) = 2 < T (s), T (s) >= 0, s J,

.

T (s) = (s) () 1, T (s) = (s) () - T (s).

T (s)

(1.3.2) k(s) := T (s)

(s). k(s) . k(s) = 0, s 1. ()

(1.3.2) k : J s 7 k(s) [0, ).

1.3.4,

.

1, . k(s) 6= 0, ,

(1.3.3) (s) :=1

k(s)

. , s J ,

(1.3.4) N(s) :=1

k(s)T (s)

(normal ve-

ctor) (s). (

1.3. 11

1.3.1) .

(1.3.4) N : J s 7 N(s) R3., s J ,

(1.3.5) B(s) := T (s)N(s)

(1.3.5) B : J s 7 B(s) R3. B(s) (binormal vector) (s). T (s), N(s) .

, s J , {T (s), N(s), B(s)}

R3,

(moving frame) Frenet (Frenet frame) (s) .

x

y

z

( )T s

( )B s

( )N s

( )T s

( )B s

( )N s

( )sb

1.3

,

{T,N,B}

12 1.

Frenet . , -

,

(s), (s) . , . -

. 1.3.

,

, . ,

s J , T (s) N(s) E. () (s) E ( T (s), N(s)) (osculating plane) (s). , (s) N(s) B(s) (normal plane) (s), (s) T (s) B(s) (rectifying plane) (s).

( )T s

( )B s

( )N s

1.4

, ( (s)) B(s), T (s) - N(s). ,

.

1.3. 13

1.3 1.4 . -

{T (s), N(s), B(s)} (s), T (s),N(s)) ...

, -

, . < N(s), B(s) >= 0, s J , - < N,B >= 0 [. (1.1.6), (1.1.8)]

< N (s), B(s) > + < N(s), B(s) >= 0.

(1.3.6) (s) := < N(s), B(s) >=< N (s), B(s) >

(torsion) (s).

(1.3.6) : J s 7 (s) R.

(1.3.6) ,

B(s), -, .

,

,

. ( -

) 1.3.5.

Frenet

, -

.

1.3.2 . u, v R3 u = 1 u v,

(u v) u = v.

. u = (a, b, c) v = (x, y, z).

u v =e1 e2 e3a b cx y z

= (bz cy)e1 (az cx)e2 + (ay bx)e3= (bz cy, cx az, ay bx).

14 1.

,

(u v) u =e1 e2 e3bz cy cx az ay bxa b c

= [c(cx az) b(ay bx)]e1 [c(bz cy) a(ay bx)]e2+

+ [b(bz cy) a(cx az)]= (c2x acz aby + b2x)e1 (cbz c2y a2y + abx)e2+

+ (b2z bcy acx+ a2z)e3 =

= ((1 a2)x acz aby, (1 b2)y abx cbz,(1 c2)z bcy acx)

= (x a(ax+ by + cz), y b(ax+ by + cz),z c(ax+ by + cz))

= (x a < u, v >, y b < u, v >, z c < u, v >)= (x, y, z) = v,

.

1.3.3 . :

T (s)N(s) = B(s),N(s)B(s) = T (s),B(s) T (s) = N(s).

. (1.3.5), B(s).

B(s) T (S) = (T (s)N(s)) T (s) = N(s).

, :

N(s)B(s) = N(s) (T (S)N(s)) == (T (S)N(s))N(s) = (N(S) T (s))N(s) = T (s).

() Frenet-Serret,

T (s), N (s), B(s) - -

. () F. Frenet J. Serret.

1.3. 15

1.3.4 . : J R3 , k > 0 {T,N,B} Frenet . ( Frenet-Serret):

(F . 1) T = kN

(F . 2) N = kT + B(F . 3) B = N .. (F. 1) N(s) [. (1.3.4)].

(F. 2) : , s J , T (s), N(s) B(s) , a, b, c : J R,

(1.3.7) N (s) = a(s)T (s) + b(s)N(s) + c(s)B(s), s J.

a, b, c (1.3.7) T (s), N(s), B(s). -

< N (s), T (s) > = a(s)< T (s), T (s) > + b(s)< N(s), T (s) >

+ c(s)< B(s), T (s) >

= a(s)1 + b(s)0 + c(s)0 = a(s).

, < T,N >= 0

< T,N >=< T , N > + < T,N >= 0,

, (F. 1),

< T,N >= < T , N >= < kN,N >= k 1 = k.

a(s) = k(s),

(1.3.7)

(1.3.8) N (s) = k(s)T (s) + b(s)N(s) + c(s)B(s), s J.

"" N(s)

< N (s), N(s) > = k(s)< T (s), N(s) > + b(s)< N(s), N(s) >+ c(s)< B(s), N(s) >

= k(s)0 + b(s)1 + c(s)0 = b(s).

16 1.

< N,N >= 1

< N,N >= 2 < N,N >= 0,

b(s) = 0,

(1.3.8)

(1.3.9) N (s) = k(s)T (s) + 0N(s) + c(s)B(s), s J.

,

< N (s), B(s) > = k(s)< T (s), B(s) > + 0< N(s), B(s) >+ c(s)< B(s), B(s) >

= k(s)0 + 0 + c(s)1 = c(s).

< B,N >= 0

< B,N >=< B, N > + < B,N >= 0,

, (1.3.6),

< B(s), N (s) >= < B(s), N(s) >= (s),

c(s) = (s),

(1.3.9)

N (s) = k(s)T (s) + (s)B(s); s J,

(F. 2).

(F. 3) B = T N , (F. 1), (F. 2) 1.3.3. ,

B = T N + T N = kN N + T (kT + B)= k 0 + (kT T + T B)= k 0 + T B= N.

1.3. 17

Frenet-Serret :

(1.3.10)

T N B

=

0 k 0k 0

0 0

.

TNB

.

,

(F. 1)(F. 3): -

( ), 2 2 ,

.

.

1.3.5 . : J R3 . :

(i) k = 0 .

(ii) k > 0, = 0 .

. (i) k = 0 T = = 0 (s) = (s) = s + [. 1.1.1(1)]. ,

, = 1.(ii) B = N = < N,B >

= 0 B = 0, B = T N , so J ,

B(s) = B(so), s J.

= 0 B(s) = B(so) s J T (s) B(so), s J < T (s), B(so) >=< (s), B(so) >= 0, s J < (s), B(so) >= 0, s J < (s), B(so) >= , s J < (s), B(so) >=< (so), B(so) >, s J < (s) (so), B(so) >= 0, s J (s) (so) B(so), s J.

(s)(so) Eo T (so) N(so), (s) Eo + (so), s J .

18 1.

, (s) E, E, Eo ( R2, 2) E 0. ,

E = (s) + Eo, s J. so J ,

(s) (so) + Eo, s J.

Eo {u, v},

(1.3.11) (s) = (so) + (s)u+ (s)v; s J,

, : J R . , - : (1.3.11)

(s) (so) = (s)u+ (s)v; s J,

,

u

< (s) (so), u > = < (s)u, u > + < (s)v, u >= (s)1 + (s)0 = (s),

=< (so), u >, . .

(1.3.11)

T (s) = (s) = (s)u+ (s)v Eo; s J,

N(s) =1

k(s)T (s) =

1

k(s)((s)u+ (s)v) Eo, s J.

, T (s) N(s) Eo B(s)

Eo. , B(s) (1.3.6), = 0.

1.3.6 . : J R3 -. k > 0.

1.3. 19

. (xo, yo) r,

(s) = r(cos

s

r, sin

s

r

)+ (xo, yo); s J [0, 2],

T (s) = (s) =( sin s

r, cos

s

r

),

T (s) = (s) =1

r

( cos sr, sin s

r

)

k(s) = T (s) = 1r.

, k > 0 .

(s) := (s) +1

kN(s).

(s) = (s) +1

kN (s)

= T (s) +1

k(kT (s) + B(s))

= T (s) T (s) + 0 = 0,

. a := (s) R3 ,

(s) a = 1kN(s) = 1

k= r,

. (s) a r. , .

1.3.7 . 1) -

[. (1.3.3)], (s) + 1kN(s).

2) 1.3.5 1.3.6

,

.

20 1.

1.4 -

Frenet - .

: I R3. , - X = (I) () . , .

1.4.1 . (. 1.2.2). T , N , B, k, - Frenet, , T,N,B, k,

T (t) := T (s(t)),( i )

N(t) := N(s(t)),( ii )

B(t) := B(s(t)),( iii )

k(t) := k(s(t)),( iv )

(t) := (s(t)).( v )

1.4.2 . : I R3 () ,

k = 3 .

. = h , h = s1 s . = s, , t I,

(t) = ( s)(t) = s(t)(s(t)) = s(t)T (s(t)),(1.4.1)(t) = s(t)T (s(t)) + s(t)2(T (s(t))

= s(t)T (s(t)) + s(t)2k(s(t))N (s(t))(1.4.2)

(t) (t) = s(t)T (s(t)) [s(t)T (s(t))++ s(t)2k(s(t))N (s(t))],

1.4. 21

(1.4.3)

(t) (t) = s(t)s(t)[T (s(t)) T (s(t))]++ s(t)3k(s(t))[T (s(t)) N(s(t))]

= 0 + s(t)3k(s(t))B(s(t))

= (t)3k(s(t))B(s(t)).

(t) (t) = (t)3k(s(t)),

k(t) := k(s(t)) =(t) (t)

(t)3 ,

.

1.4.3 . k > 0,

=< , > 2 =

[]

2.

. , - . (1.4.2)

(t) = s(t)T (s(t)) + s(t)s(t)T (s(t)) +

+[s(t)2k(s(t))]N(s(t)) + s(t)3k(s(t))N (s(t))

= s(t)T (s(t)) + s(t)s(t)k(s(t))N (s(t)) +

+[s(t)2k(s(t))]N(s(t)) s(t)3k(s(t))2T (s(t)) ++s(t)3k(s(t)) (s(t))B(s(t))

= X(t)T (s(t)) + Y (t)N(s(t)) + Z(t)B(s(t)),

X(t) = s(t) s(t)3k(s(t))2,Y (t) = s(t)s(t)k(s(t)) + [s(t)2k(s(t))],

Z(t) = s(t)3k(s(t)) (s(t)).

22 1.

(1.4.3) B(s(t)) T (s(t)) N(s(t)), :

< (t) (t) , (t) > = s(t)3k(s(t)) < B(s(t)), (t) >= s(t)3k(s(t))X(t) < B(s(t)), T (s(t)) > +

+ s(t)3k(s(t))Y (t) < B(s(t)), N (s(t)) > +

+ s(t)3k(s(t))Z(t) < B(s(t)), B(s(t)) >

= 0 + 0 + s(t)6k(s(t))2(s(t))

= (t) (t)2(s(t)),

(t) := (s(t)) =< (t) (t), (t) >

(t) (t)2 ,

.

1.5 Frenet

T (t),N(t) B(t) (. 1.4.1), , .

1.5.1 . : I R3 , , , {T (t), N(t), B(t)} Frenet.

T (t) =(t)

(t),(1.5.1)

N(t) =

((t) (t)) (t)

(t) (t) (t),(1.5.2)

B(t) =(t) (t)(t) (t) .(1.5.3)

. = h (. 1.2.2), h = s1 s .

T (t) = T (s(t)) = (s(t)) = ( h)(s(t))= (h(s(t)))h(s(t)) = (t) 1

s(t)

=(t)

(t) .

1.5. FRENET 23

, T = T s, T = T h,

N(t) = N(s(t)) =T (s(t))

k(s(t))=

(T h)(s(t))k(t)

=T (h(s(t)))h(s(t))

k(t)=

T (t)

k(t)s(t)

=T (t)

s(t)

(t)3(t) (t)

= T (t)(t)2

(t) (t) .

, T (t) N(t) . T (t) N(t), T (t) T (t). T (t) = 1., 1.3.2 u = T (t) v = T (t),

T (t) = (T (t) T (t)) T (t).

T (t) =(t)

s(t)[. (1.5.1)]

T (t) =(t)s(t) (t)s(t)

s(t)2,

T (t) =

((t)

s(t)

(t)s(t) (t)s(t)s(t)2

)

(t)

s(t)

=1

s(t)4[((t) (t)s(t) (t) (t)s(t)] (t)

=1

s(t)3[((t) (t)) (t)],

N(t) =

((t) (t)) (t)

(t) (t) (t) .

,

B(t) = B(s(t)) = T (s(t)) N(s(t)) = T (t)N(t)

=(t)

(t) ((t) (t)) (t)(t) (t) (t)

=(t)

(t) (

(t) (t)(t) (t)

(t)

(t))

=(t) (t)(t) (t) ,

24 1.

, 1.3.2,

u =(t)

(t) v =(t) (t)(t) (t) .

1.5.2 . : I R3 , , , {T,N,B} - Frenet ( ). () Frenet-Serret:

(F . 1) T = kvN ,

(F . 2) N = kvT + vB,(F . 3) B = vN , v(t) := (t) t I.. : J R3 - Frenet {T , N , B} . t I,

T (t) = (T s)(t) = T (s(t))s(t) = k(s(t))N (s(t))s(t)= k(t)v(t)N(t),

N (t) = (N s)(t) = N (s(t))s(t)= k(s(t))s(t)T (s(t)) + (s(t))s(t)B(s(t))= k(t)v(t)T (t) + (t)v(t)B(t),

B(t) = (B s)(t) = B(s(t))s(t) = (s(t))N (s(t))s(t)= (t)v(t)N(t),

.

1.6

(u1, u2, u3) (v1, v2, v3) R3.

,

.

-

R3, .

()

(e1, e2, e3) .

1.6.1 . a, b R3, (a, b, a b) .

1.6. 25

. a = (a1, a2, a3) b = (b1, b2, b3).

a b =e1 e2 e3a1 a2 a3b1 b2 b3

= (a2b3 a3b2, a3b1 a1b3, a1b2 a2b1),

a1 b1 a2b3 a3b2a2 b2 a3b1 a1b3a3 b3 a1b2 a2b1

= (a1b2 a2b1)2 + (a2b3 a3b2)2 + (a3b1 a1b3)2, . , (a, b, ab) , .

1.6.2 . : I R3 , Frenet t I R3.

: I R3 - := h : J R3. , h(t) > 0, t J . , h(t) < 0, t J , .

1.6.3 . : I = [a, b] R3 := h : [0, L()] R3 , .

, : [0, L()] R3 (s) := (L() s) - , .

. , (1.2.1),

h(s(t)) =1

s(t)=

1

(t) > 0,

t I. ,

= h ,

(s) = L() s (s) = 1, s [0, L()]. (h )(s) = h((s)) (s) < 0.

1.6.4 . Frenet,

.

26 1.

= h . (t) = ( h)(t) = (h(t))h(t),(t) = ((h(t))h(t)) = (h(t))h(t)2 + (h(t))h(t),

(t) = [(h(t))h(t)2 + (h(t))h(t)]

= h(t)3(h(t)) + 3h(t)h(t)(h(t)) + h(t)(h(t)),

(t) (t) = h(t)3((h(t)) (h(t))),((t) (t)) (t) = h(t)4(((h(t)) (h(t))) (h(t)))

< (t) (t),(t) >= h(t)6 < (h(t)) (h(t)), (h(t)) > ++ 3h(t)4h(t) < (h(t)) (h(t)), (h(t)) > ++ h(t)h(t)3 < (h(t)) (h(t)), (h(t)) >= h(t)6 < (h(t)) (h(t)), (h(t)) >.

, T(t) (t)

T(t) =(t)

(t) =( h)(t)( h)(t) =

(h(t)) h(t)(h(t)) |h(t)| = T(h(t))

h(t)

|h(t)| ,

, , (: - ) , ,

.

N(t) :

N(t) =((t) (t)) (t)(t) (t) (t)

=h(t)4

|h(t)|4 ((h(t)) (h(t))) (h(t))(h(t)) (h(t)) (h(t))

= N(h(t)),

, .

-

:

B(t) = T(t)N(t) = h(t)

|h(t)| T(h(t)) N(h(t)) =h(t)

|h(t)| B(h(t)),

1.7. 27

, , , ,

.

, k

k(t) =(t) (t)

(t)3 =|h(t)|3(h(t)) (h(t))

|h(t)|3(h(t)) = k(h(t)),

, . ,

,

(t) =< (t) (t), (t) >

(t) (t)2

=h(t)6

|h(t)|6 < (h(t)) (h(t)), (h(t)) >

(h(t)) (h(t))2= (h(t)),

.

N , k , ""

( intrinsic) .

1.7

: J R3 so J . , (so) , (so) = 0, T (so) e1 N(so) e2. B(so) = e3.

, s J , (s) = T (s),

(s) = T (s) = k(s)N(s),

(s) = (k(s)N(s)) = k(s)N(s) + k(s)N (s)

= k(s)2T (s) + k(s)N(s) + k(s)(s)B(s)., so,

(so) = 0,

(so) = T (so) = e1,

(so) = k(so)N(so) = k(so)e2,

(so) = k(so)2e1 + k(so)e2 + k(so)(so)e3.

28 1.

, Taylor,

(s) = (so) +(so)

1!(s so) +

(so)

2!(s so)2+

+(so)

3!(s so)3 +O(s3).

, -

(s) = 0 + (s so)e1 + k(so)2

(s so)2e2+

+1

6[k(so)2e1 + k(so)e2 + k(so)(so)e3](s so)3 +O(s3)

=[(s so) k(so)

2

6(s so)3

]e1+

+[k(so)

2(s so)2 + k

(so)

6(s so)3

]e2+

+[k(so)(so)

6(s so)3

]e3 +O(s

3).

(canonical representa-

tion) (so). (s so) ( 0, s so), (approximation) ( so):

(1.7.1) (s) = (s so)e1 + k(so)2

(s so)2e2 + k(so)(so)6

(s so)3e3.

,

(1.7.1) (s) =(s so, k(so)

2(s so)2, k(so)(so)

6(s so)3

).

()

so, , -

. x, y ( e1 = T (so),e2 = N(so)) x, z (: e1 = T (so),e3 = B(so)) , y, z (: e2 = N(so),e3 = B(so)) .

, x, y (: )

(s so ,k(so)2 (s so)2

), , x = s so y = k(so)2 (s so)2,

y =k(so)

2x2,

1.7. 29

[.

1.5(a)].

xx

y z

z

y

0t < 0t >

( )a ( )b

( )c

1.5

x, z (: ) (s so ,k(so)(so)6 (s so)3

). x = s so z = k(so)(so)6 (s so)3

()

z =k(so)(so)

6x3

1.5(b).

, y, z (: )

(k(so)

2 (s so)2,k(so)(so)6 (s so)3), , y = k(so)2 (s so)2

z = k(so)(so)6 (s so)3,

z2 =2

9.(so)

2

k(so)y3.

30 1.

Neil

1.5(c).

k(so) > 0 ( , Taylor

,

).

.

x

y

z

a( )a

( )b

( )c

1.6

1.7.1 . : I R3 to I , Io = (to , to + ) to, . to + t1, to + t2 Io. A := (to), B := (to + t1) := (to + t2) , P (A,B, ). P (A,B, ), B A.

1.8. 31

P (A,B, ) A - u = ~AB = (to + t1) (to) v = ~A = (to + t2) (to). Taylor

(to + t) = (to) + t(to) +

t2

2(to) +O(t

2),

t = t1, t2,

u = t1(to) +

t212(to) +O(t

21),

v = t2(to) +

t222(to) +O(t

22).

u1 :=u

t1= (to) +

t12(to) +

O(t21)

t1,

v1 :=v

t2= (to) +

t22(to) +

O(t22)

t2.

w := 2v1 u1t2 t1 =

(to) +2

t2 t1

(O(t22)

t2 O(t

21)

t1

).

u1, w u v, ,

limB,A

P (A,B, ) = limt1,t20

P (u1, w) = P(limt10

u1, limt1,t20

w)

= P((to),

(to))= P (T (to), N(to)),

, P (A,B, ) -.

1.8

, , G. W.

Leibniz, Jakob Johann Bernoulli,

L. Euler -

.

: J R3 so J , Jo so. s1, s2 Jo. A := (so), B := (s1)

32 1.

:= (s2) , C(K, r) [ P (A,B, ) (. 1.7.1)]. C(Ko, ro) , s1, s2 so.

K

r

: ( )o

sA a=

1( ) :s Ba =

2: ( )saG =

( )oT s

( )oN s

( )oB s

oK

or

1.7

, C(K, r) - P (A,B, ), C(Ko, ro) , (so).

: J R : s 7 (s) := (s)K2,

(1.8.1) (s) = 2 < (s), (s) K >= 2 < T (s), (s) K >

(1.8.2)(s) = 2 < T (s), (s) K > +2 < T (s), T (s) >

= 2 < k(s)N(s), (s) K > +2.

(so) = (s1) = (s2) = r2,

Rolle s3 so s1, s4 so s2,

(1.8.3) (s3) = (s4) = 0,

1.8. 33

(1.8.1)

< T (s3), (s3)K > = 0 = < T (s4), (s4)K >.

(1.8.4) (s3)K T (s3) (s4)K T (s4)., Rolle (1.8.3), s5 s3 s4

(s5) = 0, (1.8.2)

(1.8.5) < k(s5)N(s5), (s5)K >= 1. s1, s2 so, K Ko r ro,

s3, s4, s5 so. (1.8.4) (so)Ko T (so),

(so) Ko P (N(so), B(so)). , , (so)Ko - P (T (so), N(so)), ( ) N(so),

(1.8.6) (so)Ko = N(so), R. (1.8.5) (1.8.7) < k(so)N(so), (so)Ko >= 1. (1.8.6) (1.8.7)

< k(so)N(so), N(so) >= k(so) = 1,

= 1k(so)

.

(1.8.6)

(1.8.8) Ko = (so) +1

k(so)N(so)

(1.8.9) ro = (so)Ko = N(so) = || = 1k(so)

.

C(Ko, ro) (osculating circle) (so).

34 1.

1.8.1 . 1) (1.8.9)

(so) ,

.

2) (I) -,

N(so) =1

k(so)T (so) =

1

k(so)(so),

N(so) , .

"" .

, (1.8.8)

N(so) = k(so)(Ko (so)

)=

1

ro

(Ko (so)

),

N(so) , - (. 2).

1.9

, ,

, -

. ,

, .

(translation) c R3

c : R3 R3 : u 7 c(u) := u+ c,

(rotation) R3

,

f : R3 R3

,

< f(u), f(v) >=< u, v >; u, v R3, (

). , f , ( )

f(u v) = f(u) f(v), u, v R3.

1.9. 35

-

. : c c, ,

.

.

f c = f(c) f.

(rigid

motion).

1.9.1 . : J R3 , f c, .

(s) = f((s)) + c, s J.

, - , Frenet f .

. k, , T, N B (. k, , T, N B ) , Frenet (. )., f , / , :

T(s) = (s) = (f )(s) = f((s)) = f(T(s)),

T(s) = f(T(s)) = T(s) = 1,T (s) = (f T)(s) = f(T (s)),k(s) = T (s) = f(T (s)) = T (s) = k(s),

N(s) =T (s)

k(s)=f(T (s))

k(s)= f

(T (s)k(s)

)= f(N(s)),

B(s) = T(s)N(s) = f(T(s)) f(N(s))= f

(T(s)N(s)

)= f(B(s)),

B(s) = (f B)(s) = f(B(s)),(s) = < N(s), B(s) >= < f(N(s)), f(B(s)) >

= < N(s), B(s) >= (s),

.

1.9.2 . (f )(s) = f((s)) ( ), .

36 1.

: , :

(f )(s) = [D(f )(s)](1) = [Df((s)) Da(s)](1)= [f Da(s)](1) = f([Da(s)](1)) = f((s)).

: M f ( R3),

f((s)) = M (1(s), 2(s), 3(s))t, t .

(f )(s) = M (1(s), 2(s), 3(s))t +M (1(s), 2(s), 3(s))t= M (1(s), 2(s), 3(s))t = f((s)).

-

1.9.3 . k(s) > 0 (s), s J = [0, a], . : J R3 k . , , .

x

y

z

0( )T s

0( )B s

0( )N s

0( )T s

0( )B s

0( )N s

0( )sa

1e

2e0

3e

a

b

1.8

1.9. 37

. (1) :

T i = k Ni, N i = k Ti + Bi, Bi = Ni, i = 1, 2, 3,

T1(0) = N2(0) = B3(0) = 1,

T2(0) = T3(0) = N1(0) = N3(0) = B1(0) = B2(0) = 0.

-

,

(T1, T2, T3, N1, N2, N3, B1, B2, B3),

.

T :=

3i=1

Tiei, N :=

3i=1

Niei, B :=

3i=1

Biei.

(T,N,B)

T = kN

N = kT + BB = N

(T (0), N(0), B(0)) = (e1, e2, e3).

(T (s), N(s), B(s)) R3, s J .

x = 2ky,

y = kw kx+ z,z = kp y,w = 2ky + 2p,p = kz + q w,q = 2p,

x(0) = w(0) = q(0) = 1, y(0) = z(0) = p(0) = 0.

38 1.

x = w = q = 1, y = z = p = 0

,

x = < T, T >,

y = < T,N >,

z = < T,B >,

w = < N,N >,

p = < N,B >,

q = < B,B > .

< T, T >=< N,N >=< B,B >= 1,

< T,N >=< T,B >=< N,B >= 0,

(T (s), N(s), B(s)) R3, s J . : ,

d : J R : s 7T1(s) T2(s) T3(s)N1(s) N2(s) N3(s)B1(s) B2(s) B3(s)

, d(0) > 0. .

(s) :=

s0T (u)du :=

( s0T1(u)du,

s0T2(u)du,

s0T3(u)du

)

C2. , = T = 1, . . T = kN , N = 1, k . , B , T N , (T,N,B) , B = T N ,

B = (T N) + (T N ) =k(N N) + (k(T T ) + (T B)) = N,

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

(2) : : J = [0, a] R3 - , k ,

1.9. 39

. : {T, N, B} {T , N, B} Frenet , ,

f : R3 R3 : xT(0) + yN(0) + zB(0) 7 xT(0) + yN(0) + zB(0). f (T(0), N(0), B(0)) (T(0), N(0), B(0)), . . c := (0). g := f + c (0) = (0, 0, 0) (0), Frenet (f ) + c 0. {Tg, Ng, Bg} Frenet g = (f )+c (. 1.9.1)

< Tg, T > = < T g, T > + < Tg, T

> =

= k(< Ng, T > + < Tg, N >)

< Ng, N > = < N g, N > + < Ng, N

>

= < kTg + Bg, N > + < Ng,kT + B >)= k(< Tg, N > + < Ng, T >) +

+(< Bg, N > + < Ng, B >)

< Bg, B > = < Bg, B > + < Bg, B

>

= (< Ng, B > + < Bg, N >).

(< Tg, T > + < Ng, N > + < Bg, B >) = 0,

< Tg, T > + < Ng, N > + < Bg, B >= .

Tg(0) = f(T(0)) = T(0),

Ng(0) = f(N(0)) = N(0),

Bg(0) = f(B(0)) = B(0),

(< Tg, T > + < Ng, N > + < Bg, B >)(0) = 3,

(1.9.1) (< Tg, T > + < Ng, N > + < Bg, B >)(s) = 3, s J.

40 1.

, u, v R3, < u, v >= u v cos u v. (1.9.1) 1,

1,

cos = 0, Tg(s) = T(s), Ng(s) = N(s) Bg(s) = B(s), s J . Frenet g , (g )(0) = (0) = c

(s) =

s0T(u)du + c =

s0Tg(u)du+ c = (g )(s)

s J .

, k = k(s) > 0 = (s) (natural or intrinsic equations)

, (

).

1.10

.

.

x y.

: I R2, . = (1, 2),

T (s) := (T1(s), T2(s)) = (1(s),

2(s))

T (s) = (T1(s)2 + T2(s)2)1/2 = 1. , N(s) (

) -

,

N(s) =1

k(s)T (s) =

1

k(s)lims0

T (s+s) T (s)s

T (s), T (s+s) . , - T (s), : (T2(s), T1(s)) (T2(s),T1(s)). N(s) (. 1.9). N(s) = (T2(s), T1(s)), (T (s), N(s))

1.10. 41

R2, N(s) = (T2(s),T1(s)), .

N := (T2, T1),

(T (s), N(s)).

N = N = T

k=

T

k .

T

1T

*N

*N-

1T

2T

2T-

1T-

2T

1.9

k, - k,

k := k k = k.

,

N =T

k,

(F. 1) [. 1.3.4].

(1.10.1) T = kN.

(F. 2),

1.3.4. : (T (s), N(s)) ()

42 1.

R2, N (s) a(s), b(s) R, N (s) = a(s)T (s)+b(s)N(s), s I, N = aT +bN. ,

< N, T >= 0 < N , T > + < N, T >= 0 < aT + bN, T > + < N, kN >= 0 a+ k = 0 a = k,

< N, N >= 1 < N N >= 0 < aT + bN, N >= 0 b = 0,

(1.10.2) N = kT.

(1.10.1) (1.10.2) Frenet-

Serret .

(1.10.1) < T , N >=< kN, N >= k,

(1.10.2) k =< T, N > .

T = (1 , 2 ) N = (2, 1),

( ) :

(1.10.3) k = 1

2 12.

-

: k > 0. k = k

N =T

k=T

k= N.

,

(),

"" ,

1.10. 43

1.10.

* 0k

1.10

k < 0, k = k N = N . , , ,

"" (.

1.10).

,

,

.

1.10.1 . : I R2 ( ).

T (t) =

(1(t)

(t),2(t)

(t)),(1.10.4)

N(t) =

(

2(t)

(t),1(t)

(t)),(1.10.5)

k(t) =1(t)

2(t) 1(t)2(t)(t)3

.(1.10.6)

. = h : J R2 . (1.5.1),

T (t) =(t)

(t) =(

1(t)

(t) ,2(t)

(t)),

(1.10.4).

(1.10.5) (1.10.4) N(t) [, N(s)].

44 1.

: -

k , , (1.10.3) (iv) 1.4.1 ( k = k ),

(1.10.7) k(t) := k (s(t)) =

1(s(t))

2 (s(t)) 1 (s(t))2(s(t)),

s . i = i h (i = 1, 2),

(1.10.8) i(s(t)) = (i h)(s(t)) = i(h(s(t))) h(s(t)) =i(t)

s(t),

t I (. 1.2.2).

i s =is

i =is h.

(1.10.9)

i (s(t)) =

(is h)

(s(t))

=

(is

)(h(s(t))

) h(s(t))=

(is

)(t) 1

s(t)

=i (t) s(t) i(t) s(t)

s(t)2 1s(t)

=i (t) s(t) i(t) s(t)

s(t)3.

(1.10.8) (1.10.9) (1.10.7)

k(t) =1

s(t)4 (1(t)2(t)s(t) 1(t)2(t)s(t))

=1

s(t)3 (1(t)2(t) 1(t)2(t))

=1(t)

2(t) 1(t)2(t)(t)3 ,

.

.

1.10. 45

1.10.2 . : I R2 := h : J R2 . , () ,

,

T (t) = T(h(t)),(1.10.10)N (t) = N (h(t)),(1.10.11)k (t) = k (h(t)).(1.10.12)

. h , h < 0.

T (t) =(t)

(t) =( h)(t)( h)(t) =

=(h(t))h(t)

(h(t)) |h(t)| = (h(t))

(h(t)) = T(h(t)).

,

N (t) =( T 2 (t), T 1 (t)) = (T2 (h(t)),T1 (h(t))) = N (h(t)).

, ( h)i(t) = (i h)(t), t I i = 1, 2,

( h)i(t) = i(h(t)) h(t); t I,( h)i = (i h) h,

( h)i (t) = i (h(t)) h(t)2 + i(h(t)) h(t), t I.

, (1.10.6),

k (t) =1(t)

2 (t) 1 (t)2(t)(t)3

=( h)1(t)( h)2(t) ( h)1(t)( h)2(t)

( h)(t)3

=h(t)3

h(t)3 1(h(t))

2(h(t)) 1(h(t))2(h(t))(h(t))3

= k (h(t)),

.

-

.

46 1.

1.10.3 . : J R2 -.

(1.10.13) k = ,

(s) T (s) e1.

:

q

1e

T

y

x

1.11

. , s J , T (s) - , T (s) = (cos (s), sin (s)), : J R - .

T (s) = ( sin (s) (s), cos (s) (s)),N(s) = ( sin (s), cos (s)),

(1.10.2)

k(s) =< T(s), N(s) >=

< ((s) sin (s), (s) cos (s)), ( sin (s), cos (s)) >=(s) sin2 (s) + (s) cos2 (s) = (s),

.

, -

. ,

,

.

1.10. 47

1.10.4 . k : J = [0, a] R . : J R2, k. , : J R2 , g , g = ..

(s) :=

s0k(t)dt

(s) := (1(s), 2(s)) =

( s0

cos(u)du,

s0

sin(u)du

).

- , (1.10.3),

k.

. T T e1, (1.10.13)

(1.10.14) (s) =

s0k(t)dt + o = (s) + o,

(1.10.15)

(cos (s), sin (s)) =(cos((s) + o), sin((s) + o)

)=(cos o cos(s) sin o sin(s),cos o sin(s) + sin o cos(s)

)=

(cos o sin osin o cos o

)(cos(s)sin(s)

)= f(cos(s), sin (s)),

f , - 2 2 , o,

f(x, y) = (x cos o y sin o, x sin o + y cos o) , (x, y) R2.

,

(s) = (1(s), 2(s)) = T(s) = (cos (s), sin (s))

(s) =

( s0

cos (u)du+ xo,

s0

sin (u)du+ yo

),

48 1.

xo, yo . , (1.10.15), (s)

(s) =

( s0[cos o cos(u) sin o sin(u)]du, s

0[cos o sin(u) + sin o cos(u)]du

)+ (xo, yo) =

=

(cos o

s0

cos(u)du sin o s0

sin(u)du,

cos o

s0

sin(u)du+ sin o

s0

cos(u)du

)+ (xo, yo) =

= f

( s0

cos(u)du,

s0

sin(u)du

)+ (xo, yo) =

= f((s)) + (xo, yo),

.

1.11

1. : (i) , () P Q R3. (ii) . (iii) .(iv) .

. (i) p, q P , Q, - PQ

: [0, 1] R3 : t 7 (t) := p+ t(q p).

(ii)

(t) = q p 6= 0, ,

L() =

10(u) du =

10q pdu = q p.

, (t) = 0, 1.4.2 k(t) = 0, t [0, 1].(iii)

s(t) =

t0(u) du =

t0q p du = q pt,

1.11. 49

s : [0, 1] [0, q p] : t 7 q pt

h := s1 : [0, q p] [0, 1] : s 7 sq p ,

: [0, q p] R3

(s) := ( h)(s) = (

s

q p)

= p+s

q p(q p).

(iv) (s) =q pq p ,

L() =

qp0

(u)du = qp0

du = q p.

, k T (s) = (s) =

q pq p ,

T (s) = 0 [ (1.3.2)] k(s) = 0.

2. (0, 0) r

: [0, 2] R2 : t 7 (r cos t, r sin t).

: (i) . (ii)

. (iii)

. (iv) - k . (v) k ;

. (i) t , (t)

(t) = (r sin t, r cos t).

,

< (t), (t) >= r2 cos t sin t+ r2 cos t sin t = 0,

(t) (t).(ii) (t) = (r cos t,r sin t) = (t).

50 1.

(iii) (t) = r,

L() =

2pi0

(t)dt = 2pi0

rdt = 2r.

(iv) (t) = r 6= 0.

s : [0, 2] R : t 7 s(t) := t0(u) du = rt.

s : [0, 2] [0, 2r] ,

h := s1 : [0, 2r] [0, 2] : s sr,

: [0, 2r] R2

(s) = ( h)(s) = (sr

)=(r cos

s

r, r sin

s

r

).

T (s) = (s) =( sin s

r, cos

s

r

),

T (s) =

(1rcos

s

r,1

rsin

s

r

),

k(s) = T (s) = 1r

, (1.10.6)

k (t) =1

r3(r2 sin2 t+ r2 cos2 t) =

1

r.

3. y = x2

. (t) =(t3, t6), t R;. ,

(t) = (t, t2); t R, . ,

(t) = (1, 2t) 6= 0, t R.

1.11. 51

(t) = (0, 2) , (1.10.6)

k(t) = 2(1 + 4t2

)3/2= k(t).

, , (t) = (3t2, 6t5), t = 0 -.

4. f : R R. - T , N k .

. (t) = (t, f(t)), t R, f .

(t) = (1, f (t)) (t) = (0, f (t)).

(t) = (1 + f (t)2)1/2 6= 0; t R, , . -

, 1.10.1, -

:

T (t) =1

(t)(1(t),

2(t)

)=

(1(

1 + f (t)2)1/2 , f (t)(

1 + f (t)2)1/2

),

N(t) = (T2(t), T1(t)) =( f

(t)(1 + f (t)2

)1/2 , 1(1 + f (t)2

)1/2),

/

k(t) =1 f (t) 0 f (t)(

1 + f (t)2)3/2 = f (t)(

1 + f (t)2)3/2 .

k f (t), ( )

.

5. ,

: [0, 2] R3 : t 7 ( 12cos t, sin t,

12cos t

).

52 1.

.

T (t) = (t) =

( 1

2sin t, cos t, 1

2sin t

),

(t) =(1

2sin2 t+ cos2 t+

1

2sin2 t

)1/2= 1,

,

L() =

2pi0

(t)dt = 2pi0

1dt = 2.

T (t) = (t) =

( 1

2cos t, sin t, 1

2cos t

),

k(t) = T (t) =(1

2cos2 t+ sin2 t+

1

2cos2 t

)1/2= 1.

B(t) = T (t)N(t) = (t) (t) = 12(e1 + e3) = c

B(t) = 0, (t) = < N(t), B(t) >= 0,

, , .

6.

(t) = (a cos t, b sin t), t [0, 2], a, b > 0. ;

. x = a cos t y = b sin t,

x2

a2+y2

b2= 1,

.

:

(t) = (a sin t, b cos t) (a sin t, b cos t, 0),

1.11. 53

(t) = (a2 sin2 t+ b2 cos2 t)1/2.

(t) = (a cos t,b sin t) (a cos t,b sin t, 0)

(t) (t) = abe3,

(t) (t) = ab,

k(t) =(t) (t)

(t)3 =ab

(a2 sin2 t+ b2 cos2 t)3/2= k(t).

7. ()

: [0, 2] R3 : t 7 (r cos t, r sin t, bt); r > 0, b R.

(circular helix),

1.12. (i) . (ii) . (iii) - zOz . (iv) zOz .

. (i)

(t) = (r sin t, r cos t, b),

(t) = (r2 + b2)1/2 =: c > 0.

s : [0, 2] R : t 7 t0(u)du = tc

h : [0, 2c] [0, 2] : s 7 sc.

54 1.

2 bp

z

x

t

y

1.12

: [0, 2c] R3 : s 7 (s) = ( h)(s) =(r cos

s

c, r sin

s

c,b

cs).

(ii)

T (s) = (s) =( rcsin

s

c,r

ccos

s

c,b

c

),

T (s) =( rc2

coss

c, r

c2sin

s

c, 0)

k(s) = T (s) = rc26= 0.

, :

N(s) =1

k(s)T (s) =

( cos s

c, sin s

c, 0),

1.11. 55

B(s) =

e1 e2 e3T1 T2 T3N1 N2 N3

= (b

csin

s

c, b

ccos

s

c,r

c),

B(s) =( bc2

coss

c,b

c2sin

s

c, 0)

(s) = < N(s), B(s) >= bc2

.

(iii) ,

cos =< (t), e3 >

(t) e3 =b

c.

cos =< T (s), e3 >

T (s) e3 =< T (s), e3 > =b

c.

(iv) ,

cos =< B(s), e3 >

B(s) e3 =< B(s), e3 > =r

c.

8.

(t) :=

(t,t2

2

), 0 t 1.

. :

(t) = (1, t) (1, t, 0),(t) = (1 + t2)1/2 > 0,(t) = (0, 1) (0, 1, 0),

(t) (t) =e1 e2 e31 t 00 1 0

= e3,(t) (t) = 1.

,

k(t) =(t) (t)

(t)3 =1

(1 + t2)3/2,

56 1.

k(t) =1(t)

2(t) 1(t)2(t)(t)3 =

1 1 0 t(1 + t2)3/2

=1

(1 + t2)3/2,

k = k.

9.

: R R2 : t 7 (t2, t3). ; .

.

(t) = (2t, 3t2)

(0) = (0, 0) R. (, 0) (0,+). t 6= 0

(t) = (4t2 + 9t4)1/2,(t) = (2, 6t),

(t) (t) =e1 e2 e32t 3t2 02 6t 0

= 6t2e3 = (0, 0, 6t2),(t) (t) = 6t2,

k(t) =(t) (t)

(t)3 =6t2

(4t2 + 9t4)3/2

10. a(0) = (0, 1), .

. (t) = (cos t, sin t) (0) = (1, 0).

(t) = (cos(t + /2), sin(t + /2)) , (0) = (0, 1).

,

(t) = (t) = (cos(/2 t), sin(/2 t)), (0) = (0) = (0, 1).

11. : I R3 v R3,

(0) v (t) v, t I.

1.11. 57

(t) v, t I.

. < (0), v >= 0, < (t), v > , , < (t), v >= 0. ,

< (t), v >< , v > (t) =< (t), v > + < (t), v >=

0+ < (t), 0 >= 0.

12.

: R R3 : t 7 (3t, 3t2, 2t3)

y = 0, x = z.

. (t)

t(s) = (t) + s(t); t R,

(t) = (3, 6t, 6t2) ( s ). y = 0 x = z v = (1, 0, 1). ,

cos =< (t), v >

(t) v =3 + 6t2

9 + 36t2 + 36t4 2 =2

2= cos(

4).

13. (logarithmic spiral)

(t) = (a exp(bt) cos t, a exp(bt) sin t); t R,

a b a > 0, b < 0 (. 1.13).

limt+

(t) = limt+

(t) = (0, 0).(i) +to

(u) du .(ii)

58 1.

(iii) .

x

y

t

1.13

.

(t) = (ab exp(bt) cos t a exp(bt) sin t, ab exp(bt) sin t+ a exp(bt) cos t), ,

(t) = a(b2 + 1)1/2 exp(bt), .

(i) cos t, sin t limt+

ebt = 0,

limt+

(t) = limt+

(t) = (0, 0).

(ii) t [to,+), tto

(u) du = tto

a(b2 + 1)1/2 exp(bu)du =a

b(b2 + 1)1/2 exp(bu)

tto

=a

b(b2 + 1)1/2(exp(tb) exp(tob)),

+to

(u) du = limt+

a

b(b2 + 1)1/2(exp(tb) exp(tob)) =

= ab(b2 + 1)1/2 exp(tob),

1.11. 59

(ii).

(iii) x(t) = a exp(bt) cos t, y(t) = a exp(bt) sin t, - :

x(t) = ab exp(bt) cos t a exp(bt) sin t,y(t) = ab exp(bt) sin t+ a exp(bt) cos t,

x(t) = a(b2 1) exp(bt) cos t 2ab exp(bt) sin t,y(t) = a(b2 1) exp(bt) sin t+ 2ab exp(bt) cos t,x(t)y(t) x(t)y(t) = a2(b2 + 1) exp(2bt),x(t)2 + y(t)2 = a2(b2 + 1) exp(2bt),

[ (1.10.6) -

() ]

k(t) = |k(t)| = |x(t)y(t) x(t)y(t)|(x(t)2 + y(t)2)3/2

=1

a(b2 + 1)1/2 exp(bt).

. (i) -

, (.

1.13). (ii)

[to,+).

14. r (0, 0) xOx . (i) A, (0,0). (ii) t = 0, t = t = 2.(iii) , t [0, 2].

x

y

K K

r

A

AA

B

q

rp 2 rpO

1.14

.

60 1.

. (i) A, (0, 0), A = (x, y), K = (0, r) a K = (a, r). xOx

B AB a. AKB a = r. A A A K B KK .

x = OB AA = r r sin = r( sin ),y = OK AA = r r cos = r(1 cos ),

() =(r( sin ), r(1 cos )).

(ii)

() =(r(1 cos ), r sin ),

(0) = (0, 0), () = (2r, 0), (2) = (0, 0).

, = 0 = 2, , =

pi(s) = () + s()

=(r( sin), r(1 cos )) + s(r(1 cos ), r sin)

= (r, 2r) + s(2r, 0) = (r + 2rs, 2r).

(iii)

,

()2 = r2(1 + cos2 2 cos + sin2 )= 2r2(1 cos ) = 2r2

(1 cos2

2+ sin2

2

)= 4r2 sin2

2,

() = 2r sin

2

= 2r sin 2, 0 2.

,

L() =

2pi0

()d = 2r 2pi0

sin

2d

= 2r

pi0

2 sin d = 4r( cospi

0

)= 4r( cos + cos 0) = 8r.

1.11. 61

(2k, 2(k + 1)), k Z. .

15 (s) . (i) w (: Darboux) : T = T ,N = N B = B. (ii) T T = k2.[ , , s].

. (i) = xT + yN + zB, - x, y, z.

T = xT T + yN T + zB T= x0 yB + zN = yB + zN

, T = T , yB + zN = T = kN,

y = 0 z = k. ,

N = xT N + yN N + zB N= xB + y0 +zT = xB zT

, N = N ,xB zT = N = kT + B

x = z = k. , ,

= T + kB.

:

B = (T + kB)B = (T B) + kB B = N + 0 = B. ( ) .

(ii) T = kN

T = kN + kN = kN + k(kT + B) = k2T + kN + kB.,

T T = kN (k2T + kN + kB)= k3(N T ) + kk(N N) + k2(N B)= k3B + 0 + k2T = k2(kB + T ) = k2.

62 1.

16. : J R3 - : (s) P . .

. (s) s(t) = (s) + t(s),

t R. , ts R, s(ts) = p, p R3 P ,

(s) + ts(s) = p.

, .. (s), , - s(t) = (s) + t

(s) ts,

(s) + ts(s) = p,

s J . , s J , (s) R ( ) (s) + (s)(s) = p, .

s 7 (s) , ( ) (s)(s) = p (s)

< (s)(s), (s) >= (s) < T (s), T (s) >= (s) =.

( ) , s J :( ) (s) + (s)(s) + (s)(s) = 0

(1 + (s))T (s) + (s)k(s)N(s) = 0 1 + (s) = (s)k(s) = 0 (s) = 1 (s)k(s) = 0 (s) = c s (c s)k(s) = 0 k(s) = 0,

. c s 6= 0, (s) = 0 (). , (s) = 0 (s) = p, s J , ( ).

17. a : J R3 . P .

1.11. 63

. (s)

s(t) = (s) + tN(s); t R,

[ (s) N(s)]. , , s J , (s) R

( * ) (s) + (s)N(s) = p,

(: p P ). : ( )

(s) =< (s)N(s), N(s) >=.

( ) :

() (s) + (s)N(s) + (s)N (s) = 0 T (s) + (s)N(s) + (s)(k(s)T (s) + (s)B(s)) = 0 (1 (s)k(s))T (s) + (s)N(s) + (s)(s)B(s) = 0 1 (s)k(s) = (s) = (s)(s) = 0 = c (), k(s) = 1 (s) = 0.

6= 0, = 0, , k(s) = 1/ = .

: P .

18. a : J R3 , a .

. a (s) ( B(s))

s(t) = (s) + tB(s), t R.

, ,

s J , (s) R

( ) (s) + (s)B(s) = p.

(s) =< (s)B(s), B(s) >=,

64 1.

, ( ),

(s) + (s)B(s) + (s)B(s) = 0,

, ,

T (s) (s)(s)N(s) + (s)B(s) = 0,

1 = (s)(s) = (s) = 0,

.

19. : J R3 . 0 6= u R3 .. () : .() : , s J , (s) R

( ) (s)T (s) = u = .

(s) =< (s)T (s), T (s) >=< u, T (s) >,

(s) . ( ) :

( ) (s)T (s) + (s)T (s) = 0 (s)T (s) + (s)k(s)N(s) = 0 (s) = (s)k(s) = 0.

(s) = 0 (s) = c. c = 0, u = c T = 0, . c 6= 0, c k(s) = 0 k = 0, .

20. : I R3 (general or cylindrical helix) - 6= 0 (: ) u R3. , k > 0, :

/k = .

. . - , , u = 1. < T (s), u >

1.11. 65

= cos (.), s I, < T, u >= cos -

< T, u > (s) = 0 < T (s), u > + < T (s), u >= 0 < T (s), u >= 0 k(s) < N(s), u >= 0 < N(s), u >= 0 u N(s),

, u () T (s), B(s),

( ) u = T (s) + B(s), , R. " " T (s) B(s) (. 1.3.4)

=< u, T (s) >= cos =< u,B(s) >= cos(2 ) = sin .

( ) u = cos T (s) + sin B(s),

0 = cos T (s) + sin B(s)

= cos k(s)N(s) sin (s)N(s)= (cos k(s) sin (s))N(s),

cos k(s) sin (s) = 0

( ) (s)k(s)

=cos

sin = cot = c ().

, (s)k(s)

= c. cot

(,+), c = cot . , ( ), :

k(s) cos = (s) sin k(s) cos N(s) (s) sin N(s) = 0 cos T (s) + sin B(s) = 0 cos T (s) + sin B(s) = u = . R3 cos < T (s), T (s) > +sin < B(s), T (s) >=< u, T (s) >cos =< u, T (s) >= .

66 1.

: 1) , -

, -

1.5.2. v .2) u ,

v = u/u, T (s).3) (iii) 7

.

21. .

P .

. , .

k > 0. = 0. (s)

(s) + t1T (s) + t2N(s), t1, t2 R. P , s J , (s), (s) R, ( ) (s) + (s)T (s) + (s)N(s) = p(: p P ).

, : J R . , ( )

(s)T (s) + (s)N(s) = p (s),

(s) =< (s)T (s), T (s) > + < (s)N(s), T (s) >=,(s) =< (s)T (s), N(s) > + < (s)N(s), N(s) >=,

, . ( ) :(s) + (s)T (s) + (s)T (s) + (s)N(s) + (s)N (s) = 0T (s) + (s)T (s) + (s)k(s)N(s) + (s)N(s)

(s)k(s)T (s) + (s)(s)B(s) = 0(1 + (s) (s)k(s))T (s) + ((s)k(s) + (s))N(s) + (s)(s)B(s) = 0,

1 + (s) (s)k(s) = 0,(s)k(s) + (s) = 0,

(s)(s) = 0.

1.11. 67

= 0, = 0. = 0,

1 + (s) = 0 (s)k(s) = 0,

(s) = c s (c s)k(s) = 0, s J, k = 0, . , = 0, . [, c s 6= 0, , ( ), (s) = p, s ().]

.

22.

.

. (i) (s) . (x, y, z) R3 (so) (x, y, z) (so) N(so) B(so), T (so) =

(so). (x, y, z) (so)

( ) < (x, y, z) (so), T (so) >= 0,

, ,

< (x, y, z) (so), (so) >= 0.(ii) (t), - . (x, y, z) (to) - ()

< (x, y, z) (to), T (to) >= 0.

(1.5.1),

< (x, y, z) (to), (to)

(to) >= 0,

, ,

< (x, y, z) (to), (to) >= 0.

.

(t) = (a sin2 t, a sin t cos t, a cos t)

(0, 0, 0).

68 1.

. , (0, 0, 0) (t),

< (0, 0, 0) (t), (t) >=< (t), (t) >= 0, t R.

(t) =(2a sin t cos t, a(cos2 t sin2 t),a sin t).

,

< (t), (t) >=

< (a sin2 t, a sin t cos t, a cos t), (2a sin t cos t, a(cos2 t sin2 t),a sin t) >=2a2 sin3 t cos t+ a2 sin t cos3 t a2 sin3 t cos t a2 sin t cos t =

a2 sin3 t cos t+ a2 sin t cos3 t a2 sin t cos t =a2 sin t cos t(sin2 t+ cos2 t) a2 sin t cos t = 0.

23.

.

. (i) (s) . (x, y, z) R3 (so) (x, y, z) (so) T (so) N(so), B(so). (x, y, z) (so)

( ) < (x, y, z) (so), B(so) >= 0.

B(so) = T (so)N(so) = (so) T(so)

k(so)= (so)

(so)

k(so),

( )

< (x, y, z) (so), (so) (so) >= 0.

(ii) (t), - . (x, y, z) (to) ( )

< (x, y, z) (to), B(to) >= 0.

1.11. 69

[. (1.5.3)]

B(to) =(to) (to)(to) (to)

< (x, y, z) (to), (to) (to)

(to) (to) >= 0,

, ,

< (x, y, z) (to), (to) (to) >= 0.

. (t) = (cos t, sin t, t), t R. () (t) (to),

to =

2.

. -

< (x, y, z) (2

),

(2

)

(2

)>= 0.

:

(2

)=(cos

2, sin

2,

2

)=(0, 1,

2

)(t) = ( sin t, cos t, 1)(2

)=( sin

2, cos

2, 1)= (1, 0, 1)

(t) = ( cos t, sin t, 0)(2

)=( cos

2, sin

2, 0)= (0,1, 0)

< (x, y, z) (0, 1, 2), (1, 0, 1) (0,1, 0) >= 0

< (x, y 1, z 2), (1, 0, 1) (0,1, 0) >= 0

x y 1 z pi21 0 10 1 0

= 0 x+ z =

2.

70 1.

24. -

.

. (i) (s) . (x, y, z) R3 (so) - (x, y, z) (so) T (so) B(so), N(so). (x, y, z) (so)

( ) < (x, y, z) (so), N(so) >= 0.

N(so) =T (so)

k(so)=(so)

k(so),

( )

< (x, y, z) (so), (so)

k(so)>= 0,

, ,

< (x, y, z) (so), (so) >= 0.(ii) (t), - . (x, y, z) (to) ( )

< (x, y, z) (to), N(to) >= 0. [. (1.5.2)]

N(to) =

((to) (to)

) (to)(to) (to) (to) ,

< (x, y, z) (to),((to) (to)

) (to)(to) (to) (to) >= 0,

, ,

( ) < (x, y, z) (to), ((to) (to)) (to) >= 0.

. (t) =(t,t2

2,t3

3

), t R.

(t) to = 1.

1.11. 71

. ( ),

< (x, y, z) (1), ((1) (1)) (1) >= 0.:

(1) =(1,

1

2,1

3

)(x, y, z) (1) = (x 1, y 1

2, z 1

3

)(t) =

(1, t, t2

)(1) = (1, 1, 1)

(t) = (0, 1, 2t)

(1) = (0, 1, 2)

(1) (1) =e1 e2 e31 1 10 1 2

= (1,2, 1)

< (x 1, y 1/2, z 1/3), (1,2, 1) (1, 1, 1) >=e1 e2 e31 2 11 1 1

= 0,

x z = 23.

25. : J R3 , p R, . k 6= 0 , :

p = 1kN

(1k

) 1B,(I)

R2 =(1k

)2+

((1k

) 1

)2,(II)

k=

((1k

) 1

).(III)

72 1.

.

p = R < p, p >= R2 < p, p > (s) = 0 2 < , p > (s) = 0 < T, p > (s) = 0 ((s) p) T (s).

, s J , (s) p N(s), B(s) (: ), , : J R, ( ) (s) p = (s)N(s) + (s)B(s). < T, p >= 0, :

< T, p > (s) = 0 < T (s), (s) p > + < T (s), (s) >= 0 < k(s)N(s), (s) p >= < T (s), T (s) >= 1 < k(s)N(s), (s)N(s) + (s)B(s) >= 1

k(s)(s) < N(s), N(s) > +k(s)(s) < N(s), B(s) >= 1 k(s)(s) = 1 (s) = 1

k(s).

(s) ( ) ( - ) :

() (s) p = 1k(s)

N(s) + (s)B(s)

(s) = ( 1k(s)

)N(s) 1

k(s)N (s) + (s)B(s) + (s)B(s)

T (s) = ( 1k(s)

)N(s) 1

k(s)

( k(s)T (s) + (s)B(s))+ (s)B(s) (s)N(s)

T (s) = T (s) +(( 1k(s)

) (s)(s))N(s) + ((s) (s)k(s)

)B(s)

(( 1k(s)

) (s)(s)

)N(s) +

((s) (s)

k(s)

)B(s) = 0

(s)(s) = ( 1k(s)

) (s) =

(s)

k(s).

1.11. 73

( k 6= 0, )

(s) = ( 1k(s)

) 1(s)

.

( ) (s) (s), , (I).

(II)

R2 =< p, p >

=< 1kN

(1k

) 1B,1

kN

(1k

) 1B >

=1

k2< N,N > +2

1

k

(1k

) 1< N,B > +

((1k

))2 12

< B,B >

=1

k2+((1

k

))2 12

,

( II).

(II) :

(II) 21k

(1k

)+ 2

(1k

)(1

)((1

k

)(1

))= 0

1k= 1

((1k

)(1

)),

(III).

26. k, 6= 0. (1k

) 6= 0 (1k

)2+

((1k

) 1

)2= ,

.

. (1k

)2+

((1k

) 1

)2= c2, (II)

25. , (II) (III) ,

k+

((1k

) 1

)= 0.

(I) ,

p := +1

kN +

(1k

) 1B,

74 1.

p s. 1

kN +

(1k

) 1B

((1k

)2+((1

k

) 1

)2)1/2= c.

p = 0, p , p c. ,

p = +(1k

)N +

(1k

)N +

((1k

) 1

)B +

(1k

) 1B

= T +(1k

)N +

1

k(kT + B) +

((1k

) 1

)B +

(1k

) 1(N)

= T +(1k

)N T +

kB +

((1k

) 1

)B

(1k

)N

= T +(1k

)N T +

kB +

((1k

) 1

)B

(1k

)N

= 0T + 0N +(k+((1

k

) 1

))B = 0

.

. ,

(. -

)

(1k

)2+

((1k

) 1

)2= c.

27. : I R3 , k(s) > 0 s I, (s). (s) :=T (s), T (s) . (i) . (ii) ( so),

(s) = k(s), s I. (iii) k .

. (i) (s) = T (s) = k(s)N(s). , (s) =k(s) 6= 0, .(ii)

(s) =

sso

(u) du,

1.11. 75

(s) = (s) = T (s) = k(s)N(s) = k(s).

(iii) ( k = 1, ),

1.4.2 1.4.3 .

( s):

= T = kN,

= kN + kN

= kN + k(kT + B)= k2T + kN + kB,

= kN (k2 + kN + kB)= k3N T + kkN N + k2N B= k2T + 0 + k3B = k2T + 0N + k3B,

2 = k6 + k42 = k4(k2 + 2).,

k = 3 =

k2(k2 + 2

)1/2k3

=

(k2 + 2

)1/2k

.

. - , , :

= 2kkT k2T + kN + kN + kB + k B + kB= 2kkT k3N + kN + k(kT + B) + (k + k )B + k(N)= 3kkT + (k3 + k k2)N + (2k + k )B.

,

=< , > 2 =

k3(k k)k4(k2 + 2)

=k kk(k2 + 2)

.

. (spherical indi-catrix) () T .

76 1.

N B. R3 0.

28.

(t) =(et cos t, et sin t

), t R.

(i) limt+ (t) = limt+ (t) = (0, 0). (ii) -

to(u) du. (iii) .

. (.

13, a = 1 b = 1). (i) . (ii) :

(t) =( et cos t et sin t,et sin t+ et cos t),

(t) = ((et cos t et sin t)2 + (et sin t+ et cos t)2)1/2 = 2et, tto

(u)du =2

tto

eudu =2( eu)t

to=2(eto et).

, to

(u)du = limt+

tto

(u) du =2 eto .

(iii)

1(t) := et cos t , 2(t) := e

t sin t,

:

1(t) = et cos t et sin t,2(t) = et sin t+ et cos t,1(t) = 2e

t sin t,

2(t) = 2et cos t,1(t)

2(t) 1(t)2(t) = 2e2t,

(t)2 = 1(t)2 + 2(t)2 = 2e2t,

k(t) = |k(t)| = |1(t)

2(t) 1(t)2(t)|(t)3 =

12et.

[,

() 13,

a = 1 b = 1.]

1.11. 77

29. . ( ) k = = . ;

. > 0 6= 0. - , ,

. 7

(s) =(r cos

s

c, r sin

s

c,b

cs),

c2 = r2 + b2. ( ),

r

r2 + b2=

b

r2 + b2,

r b -

r =

2 + 2 b =

2 + 2.

,

(s) =

(

2 + 2cos

(s2 + 2

),

2 + 2sin(s2 + 2

),2 + 2

2 + 2s

),

-

k = = ( - r b). , , , .

30. : I R2 k(s) < 1, s I.

(s) := (s) +N(s), s I.

( ), N(s) . - k

k =k

1 k .

78 1.

. () :

(s) = (s) +N (s) = T (s) k(s) T (s) = (1 k(s)) T (s) 6= 0.() ,

1.4.2.

s,

= (1 k)T = |1 k| = 1 k = (1 k)T + (1 k)T = (1 k)T + (1 k)kN

= (1 k)T ((1 k)T + (1 k)kN= (1 k)(1 k)T T + (1 k)2kT N= k(1 k)B

= k(1 k)2

k =k(1 k)2(1 k)3 =

k

1 k .

31. (s) (s) 6= 0. k B(s).

. Frenet-Serret B,

N(s) = B(s)

(s),

:

k(s) = T (s) = (N(s)B(s))

= ( B(s)(s)

B(s))= ( B(s)

(s)

) B(s) + ( B(s)(s)

)B(s)= B

(s)(s) +B(s) (s)

(s)2B(s) + 0

=1

(s)2( (s)B(s) (s)B(s))B(s).

32.

, .

1.11. 79

. , . 16-19

21,

(*) p = (s) + (s)(N(s) +B(s)),

p (s) - . (*) , s,

T + (N +B) + (kT + B N) = 0,

, ,

(1 k)T + ( )N + ( + )B = 0,

1 k = 0, = 0, + = 0. = 0, = c =

1 ck = 0, c = 0.

c 6= 0 k = 1/c , = 0. c.

33. (s), s J , , B(s), s J . , > 0.

. B = N , :

= | | N = N = B,

N = B

.

= T = N B

, T :

=

T + c.

2

2.0

R3 ( ,

..), "" R2,

. .

[] , ,

.

. -

.

. ,

-

() ,

, Mercator, Lambert,

.. ,

. ,

( -

) [8, . 116130]

[9, . 124159]. (. . 536542)

, .

-

( 1

).

L. Euler, G. Monge,

81

82 2.

C. F. Gauss. , 1827

Gottingen Disquisitiones

Generales circa Superficies Curvas (" -

"),

. ,

,

O. Bonnet, E. B. Christoffel, D. Codazzi, G. Darboux, Ch. Dupin, T. Levi-

Civita`, J. B. Meusnier, F. Mindig, A. F. Mobius, J. Plucker, O. Rodrigues,

H. Poincare, J. Weingarten.

, Gauss, -

-

,

Egregium (" ")

" ", R3. -

-

,

( ).

B. Riemann -

,

( ).

2.1

2.1.1 . (surface parametrization)

2- (coordinate system) 2-

(chart or patch), (parametrized

surgace) (U, r,W ), U R2 , r : U R3 - W = r(U), :

(1) r : U W = r(U) (2) q = (u, v) U , Dr(q) : R2 R3 11. 2.1.

2.1.2 . 1) (1), W R3.

2) r : U R2 R3,

r = (x, y, z)

x := pr1 r, y := pr2 r, z := pr3 r : U R r.

r(u, v) =((x(u, v), y(u, v), z(u, v)

), (u, v) U.

2.1. 83

urvr

ou

ou

UWr

q

x

y

z

u

v

2.1

r , , . , q = (uo, vo) U , Jacobi (. , .15)

(2.1.1) Jqr =

x

u

q

x

v

q

y

u

q

y

v

q

z

u

q

z

v

q

.

, (2) 2.1.1 Jqr 2,

(x, y)

(u, v)

q

:= det

x

u

q

x

v

q

y

u

q

y

v

q

, (x, z)(u, v)

q

:= det

x

u

q

x

v

q

z

u

q

z

v

q

,

(y, z)

(u, v)

q

:= det

y

u

q

y

v

q

z

u

q

z

v

q

,

, ,

(2.1.2)

(x

u

q

,y

u

q

,z

u

q

)(x

v

q

,y

v

q

,z

v

q

)6= 0.

84 2.

3) -

(2.1.3) ru(q) :=r

u

q

=

(x

u

q

,y

u

q

,z

u

q

)

(2.1.4) rv(q) :=r

v

q

(x

v

q

,y

v

q

,z

v

q

)

, ()

R3, r(q) r(U) r(q). 2.6.

4) , (2.1.2) -

(2.1.2) ru(q) rv(q) 6= 0.

,

.

5) , r, U W . - (), -

, , .

2.1.3 . (U, r,W ) . :

(1) r .

(2) r u v.

, r(U) , .

. (1) r = c, Dr(q) = 0, q U , .(2) r u, ru(q) = 0, ru(q) rv(q) = 0, . v.

(U, r,W ). - ru(q) rv(q), q U .

qo = (uo, vo) U

(2.1.5) : u (u) := r(u, vo) = (x(u, vo), y(u, vo), z(u, vo)) .

2.1. 85

W = r(U),

(2.1.6) (uo) =r

u

qo

= ru(qo).

ru(qo) =ru

qo

, uo,

u 7 (u) := r(u, vo). , rv(qo) =rv

qo

, vo,

v 7 (v) := r(uo, v). q = (u, v) U , W = r(U)

: u r(u, v) : v r(u, v),

ru(q) =r

u

q

rv(q) =r

v

q

,

q = (u, v) U (. 2.1). , ( -

) (coordinate curves) -

(parameter curves) (U, r,W ).

2.1.4 . (regular surface) -

S R3 {(Ui, ri,Wi)}iI 2- (), Wi S ( S )

S =iI

Wi.

, p S, 2- (Up, rp,Wp), p Wp Wp S . 2- S (2-) .

( S, R3) W S S, W = A S, A R3 .

(U, r,W ) - , S = W , . W , W = W R3.

86 2.

2.2

1. R2.

U R2 . i : U S = U {0} R3 : (u, v) 7 (u, v, 0)

11 ,

i1 : U {0} U : (u, v, 0) 7 (u, v)( )

Jqi =

1 00 10 0

2, (u, v) U . (U, i, U {0}) 2- S = U {0} . U {0} , . U U {0}( i) U .

2. R3.

:= {(x, y, z) R3 : Ax+By + Cz +D = 0}. 2- : (A,B,C) 6= (0, 0, 0), C 6= 0,

r : R2 R3 : (u, v) 7(u, v,

D AuBvC

).

r 11, . -

r1 : R2 : (x, y, z) 7 (x, y). r1 ,

R3 R2 : (x, y, z) 7 (x, y)

, r : R2 . , r -

r = (x, y, z),

2.2. 87

x(u, v) = pr1(u, v) = u,

y(u, v) = pr2(u, v) = v,

z(u, v) =D AuBv

C,

J(u,v)r =

1 00 1

AC

BC

2. 2.1.1. -

, .

3. S2.

S2 = {(x, y, z) R3 : x2 + y2 + z2 = 1}

R3

U R2. , , , .

, , -

, .

r+z : D(0, 1) R3 : (u, v) 7(u, v,

1 u2 v2),

D(0, 1) :={(u, v) R2 : u2 + v2 < 1}.

( , )u v

2 21( , , )u vu v - -

(0,1)D

S z+

2.2

88 2.

r+z 11

S+z :={(x, y, z) S2 : z > 0},

S2, S+z = S2 (RR (0,+)).

r+z

(r+z )1 : S+z D(0, 1): (x, y, z) 7 (x, y).

r+z (r+z )

1 , , r+z -. (u, v) D(0, 1)

J(u,v)r+z =

1 00 1

u1 u2 v2

, v1 u2 v2

,

r+z () 2 ( Jacobi 2), (u, v) D(0, 1).

, S2

Sz :={(x, y, z) S2 : z < 0},

S+x :={(x, y, z) S2 : x > 0},

Sx :={(x, y, z) S2 : x < 0},

S+y :={(x, y, z) S2 : y > 0},

Sy :={(x, y, z) S2 : y < 0},

r+x : D(0, 1) S+x : (u, v) 7(

1 u2 v2, u, v),rx : D(0, 1) Sx : (u, v) 7

(1 u2 v2, u, v),r+y : D(0, 1) S+y : (u, v) 7

(u,1 u2 v2, v),

ry : D(0, 1) Sy : (u, v) 7(u,1 u2 v2, v),

rz : D(0, 1) Sz : (u, v) 7(u, v,1 u2 v2).

6 -

, . -

2.3.

2.2. 89

S+

z

S-

z

S+

x

S-

x

S+

y

S-

y

z

x y

2.3

S2, - ,

.

.

N E (. 2.4). P , N , NP PN E. , N , , 11

r1N : S2 \ {N} R2 : (x, y, z) 7

(x

1 z ,y

1 z),

90 2.

. -

r1N

rN : R2 S2 \ {N} : (u, v) 7

(2u

1 + u2 + v2,

2v

1 + u2 + v2,1 + u2 + v21 + u2 + v2

).

N

S

x

y

z

P

NP

E

0

2.4

rN , rN : R

2 S2 \ {N} . rN (u, v) R2

J(u,v)rN =2

(1 + u2 + v2)2

1 u2 + v2 2uv2uv 1 + u2 v2

2u 2v

,

() 2, (R2, rN , S

2\{N}) 2- () S2.

( -

r1N , rN , R2.)

. 2.1.2(4),

J(u,v)rN (rN )u(q) (rN )v(q) : ,

(*) (rN )u(q) (rN )v(q) = 4(1 + u2 + v2)2

6= 0,

2.2. 91

q = (u, v) R2. .,

rS : R2 S2 \ {S} : (u, v) 7

(2u

1 + u2 + v2,

2v

1 + u2 + v2,1 u2 v21 + u2 + v2

)

2- (R2, rS , S

2 \ {S}), S := (0, 0,1) S2. rS

r1S : S2 \ {S} R2 : (x, y, z) 7

(x

1 + z,

y

1 + z

),

( ).

S2 ., S2 \ {N} S2 \ {S} S2 ( ),

S2 \ {N} = S2 (R3 \ {N}), S2 \ {S} = S2 (R3 \ {S}).

,

, -

(geographical coordinates),

7 3.

4. C- f : U R2 R. U R2 f : U R. , f

f :={(u, v, f(u, v)) : (u, v) U}.

r : U f : (u, v) 7 r(u, v) := (u, v, f(u, v)). r - : x = pr1, y = pr2, z = f . r 11 ( f )

:= r1 : f U : (u, v, f(u, v)) 7 (u, v), , . r . ,

J(u,v)f =

1 00 1

f

u

(u,v)

f

v

(u,v)

,

92 2.

Jacobi r 2. , (U, r,f ) f , -, .

r r(u, v) = (u, v, f(u, v)). , {(u, v, f(u, v))} f : R2 R f(x, y) = z. , g : R2 R g(x, z) = y {(u, g(u, v), v)}, r(u, v) = (u, g(u, v), v), h : R2 R h(y, z) = x {(h(u, v), u, , v)}, r(u, v) = (h(u, v), u, v).

() Monge. -

( f ). .

.

. :

f : V R (V R3 ),

(2.2.1) fx(p) :=f

x

p

, fy(p) :=f

y

p

, fz(p) :=f

z

p

,

p V.2.2.1 . V R3 , f : V R c f(V ) : p f1(c) f1({c}), Df(p) : R3 R . S := f1(c) .

, ,

Df(p). p = (x, y, z), , (2.2.1), Jacobi f p

Jpf =(fx(p), fy(p), fz(p)

).

Df(p) . t R t 6= 0, (a, b, c) R3, (2.2.2) Df(p)(a, b, c) = afx(p) + bfy(p) + cfz(p) = t 6= 0. ,

.

, , , fz(p) 6= 0. , t R, (0, 0, t fz(p)1) R3

Df(p)(0, 0, t fz(p)1

)= 0 fx(p) + 0 fy(p) + t fz(p)1 fz(p) = t.

2.2. 93

,

Jacobi,

Df(p) 6= 0. ,

Df(p) Df(p) 6= 0 (fx(p), fy(p), fz(p)) 6= 0.

.

. po = (xo, yo, zo) S. (fx(po), fy(po), fz(po)

) 6= 0. ( ) fz(po) 6= 0. f f : V R2 R R,

Uo R2 Ao R (xo, yo) zo , Uo Ao V , g : Uo Ao, g(xo, yo) = zo f(x, y, g(x, y)) = c, (x, y) Uo. , :{

(x, y) Uoz = g(x, y) Ao

} (x, y, z) (Uo A) f1(c).

ro : Uo R3 : (u, v) 7 (u, v, g(u, v)).

Wo := ro(Uo) = (Uo Ao) f1(c) = (Uo Ao) S, S (. 2.1.4). , Wo g, Wo =g. (Uo, ro,Wo) Monge S po Wo. p S, S (Monge), S (, , W ).

5 ( 2.2.1).

(i) S2:

f : R3 R : (x, y, z) 7 x2 + y2 + z2.

94 2.

S2 = f1(1). po = (xo, yo, zo) f1(1).

Jpof =

(f

x

po

,f

y

po

,f

z

po

)= 2(xo, yo, zo) 6= (0, 0, 0),

2.2.1, S2 .

(ii) , -

[ abc 6= 0]:

x2

a2+y2

b2+z2

c2= 1 ()()

x2

a2+y2

b2 z

2

c2= 1 ( )()

x2

a2 y

2

b2 z

2

c2= 1 ( )()

x2

a2+y2

b2 z = 0 ( )()

x2

a2 y

2

b2 z = 0 ( )()

x2

a2+y2

b2 z

2

c2= 0, (x, y, z) 6= (0, 0, 0) ( )()

2.5()2.5() -

. : ellipsoid,

hyperboloid (of one sheet), hyperboloid (of two sheets), elliptic paraboloid,

hyperbolic paraboloid, quadratic cone.

x

y

z

2.5()

2.2. 95

z

x

y

2.5()

z

x

y

2.5()

96 2.

x

y

z

2.5()

x

y

z

2.5()

2.3. 97

x

y

z

2.5()

2.3

() Monge

.

2.3.1 . S . po S Wo po S, Wo - f : Vo R2 R. , , () Monge

.

. S , (U, r, r(U)) po r(U) r(U) S . , 2.1.2,

(x, y)

(u, v)

q

,(x, z)

(u, v)

q

,(y, z)

(u, v)

q

, q U . po = (xo, yo, zo), qo := r1(po) , ,

(x, y)

(u, v)

qo

6= 0.

98 2.

( 1 2 )

: R3 R2 : (a, b, c) 7 (a, b)

r,

r : U R2 : (u, v) 7 (x(u, v), y(u, v)).

R2 U r - S R3

R2

?

r-

2.1

Jacobi r qo

det (Jqo( r)) =x

u

qo

x

v

qo

y

u

qo

y

v

qo

=(x, y)

(u, v)

qo

6= 0,

( , , D( r)(qo) ). , ,

r qo, Uo U qo Vo ( r)(U) R2 ( r)(qo) = (xo, yo),

r|Uo : Uo Vo . Wo := r(Uo). Wo - S, r Uo .

Wo = f , f . (. 2.6)

f := z ( r|Uo)1 : Vo R

( z = pr3 r). Wo = f . ,

p Wo = r(Uo) p = r(u, v), (u, v) Uo.

(u, v) Uo (a, b) Vo (a, b) = ( r|Uo)(u, v) = ( r)(u, v),

2.3. 99

( )r UU

oW

r

1( )oU

f z rp-

= o o

zor

p

S

1( )oU

rp-

o

oUrp o

oU

oq

op

( , )o o

x y

2

2

( ( ))r Up

oV

2.6

Wo p = r(u, v) = (x(u, v), y(u, v), z(u, v))= (( r)(u, v), z(u, v)) = (a, b, z ( r|Uo)1(a, b))=(a, b, f(a, b)

) f . (. 4 2.2), (Vo, ro,Wo), ro(a, b) = (a, b, f(a, b)), (a, b) Vo, Monge po Wo. ro r1o = |Wo .2.3.2 . S . U R2 - r : U R3 11 , r(U) S Dr(q) 11, q U . r1 : r(U) U , r (U, r, r(U)) S.

. qo U po = r(qo) r(U). , ,

(x,y)(u,v)

qo6= 0. -

, r|Uo : Uo Vo,

100 2.

, , . qo Uo,Uo U Wo = r(Uo), Monge (Vo, ro,Wo) |Wo = r1o (, , ro -).

r|Uo =(ro |Wo

) r|Uo = ro ( r|Uo), r|Uo . - r qo, qo. , r , , -

.

,

S , (U, r,W ) S, r .

2.3.3 . S (Ui, ri, ri(Ui)), i = 1, 2, S. W := r1(U1) r2(U2) 6= , - (change of coordinates)

() (transition map)

U1 r11 (W )h:=r1

2r1 r12 (W ) U2

(. 2.7).

. q1 r11 (W ) q2 := h(q1) r12 (W ). h q1.

(U2, r2, r2(U2)) , -

, , (x2,y2)(u,v)

q26= 0.

Monge (Vo, ro,Wo) - 2.3.1. Wo = f , f : Vo R

r1o = |Wo : Wo Vo : (x, y, z) 7 (x, y).

, Uo U2 q2 Uo

r2|Uo : Uo Vo

.

102 2.

2.4

R3 : I (I R ), C = (I) . , C.

, C - S R3 (surface ofrevolution). C (profile curve) (generating curve) (axis) S.

C, , , - S (parallels of S), C ( )

(meridians) S.

xOz zOz. , C zOz, , .. x > 0. (v) =(x(v), z(v)), v I. 2.8.

vo I po = (vo) = (x(vo), z(vo)) (x(vo), 0, z(vo)). , xOz z, po x(vo) xOy, z(vo). (x(vo) cos u, x(vo) sinu, z(vo)), u [0, 2).

z

y

x

( )o

z v

( )o

x v

( )o o

v pa =

c

2.8

2.4. 103

S

S ={(x(v) cos u, x(v) sin u, z(v)) : (u, v) [0, 2) I}.

.

[

() , , -

]:

vo Io I, vo Io (Io) g : Io R, (Io) = g ., , (vo) = (x

(vo), z(vo)) 6= 0,

.

x(vo) 6= 0. , , Io I Jo R, vo Io x|Io : Io Jo .

g : = z (x|Io)1 : Jo R.

p = (x(v), z(v)) (Io), v Io, t Jo x(v) = t. ,

(Io) p = (x(v), z(v)) =(t, (z x|Io

)(t) = (t, gt)) g

.

,

z(v) = g(t) = g(x(v)), . z = g(x).

z = g(x) f(x, z) = c () Df(x, z) 6= 0.

, z = g(x) f(x, z) := g(x) z, f(x, z) = 0 Df(x, z) J(x,z)f = (g(x), 1) 6= 0 (. 2.2.1). :

f(x, z) = c Df(x, z) 6= 0, (fx, fz) 6= 0. , , fz 6= 0, , ,z = g(x) .

, ( -

) C f(x, z) = c, Df(x, z) 6= 0, (x, z) C. - , ,

, v I,

104 2.

, Io, . ,

(x, y, z) S x2 + y2 = x(v)2 z = z(v) , v I f(x2 + y2, z) = c.

h : R R R R : (x, y, z) 7 f(x2 + y2, z).

S = g1(c).

2.2.1, S - : p = (x, y, z) h1(c), Dh(p) ( Dh(p) 6= 0), Df(x, z) 6= 0.

, h = f , (x, y, z) =(x2 + y2, z

). , (a, b) = (p), -

Jacobi,

Dh(p) Jph =(J(a,b)f

) (Jp)=(g(a), 1

) (x(x2 + y2) 12 y(x2 + y2) 12 00 0 1

)

=(g(a)x(x2 + y2)

1

2 , g(a)y(x2 + y2)1

2 , 1)6= 0.

S , .

(U, r,W ), U = (0, 2) I, r

r : U R3 : (u, v) 7 (x(v) cos u, x(v) sin u, z(v))

W = r(U). U R2 r ( ).

r 11: r(u1, v1) = r(u2, v2), z(v1) = z(v2) x(v1)

2 = x(v2)2. x > 0,

x(v1) = x(v2),

(v1) = (x(v1), z(v1)) = (x(v2), z(v2)) = (v2).

C , 11, v1 = v2 = v, , r(u1, v) = r(u2, v), u1, u2 (0, 2) sin cos, u1 = u2.

2.4. 105

Dr(q) 11, q = (u, v) U . ,

ru(q) = (x(v) sin u, x(v) cos u, 0),rv(q) = (x

(v) cos u, x(v) sin u, z(v)),

ru(q) rv(q) = e1 e2 e3x(v) sin u x(v) cos u 0

x(v) cos u x(v) sin u z(v)

= (x(v)z(v) cos u, x(v)z(v) sin u, x(v)x(v))= x(v)(z(v) cos u, z(v) sin u, x(v)),

ru(q) rv(q) = x(v) (x(v)2 + z(v)2

)1/2.

x(v) > 0, ru(q) rv(q) = 0 (v) = (x(v)2+ z(v)2)1/2 = 0, v, , C . ru(q)rv(q) 6= 0 Dr(q) 11.

W = r(U) S, W S . (U, r,W ) 2.3.2, S. u , , v ().

u, .. U =(, ) I r(u, v) = (x(v) cos u, x(v) sin u, z(v)), S, S.

, -

(

),

z ( , ), -

. ,

,

. -

, .

, .

106 2.

: ( - torus) T .

z ( xOz) (a, 0, 0) , 0 < < a.

(v) = (x(v), z(v)) = (a+ cos v, sin v) (a+ cos v, 0, sin v).

T ={(

(a+ cos v) cos u, (a+ cos v) sin u, sin v) (u, v) [0, 2) [0, 2)}.

x

y

z

a

r

2.9

r(u, v) =((a+ cos v) cos u, (a+ cos v) sin u, sin v

),

(u, v) (0, 2) (0, 2), T , "" () "" (). T,

, , (, ) (, ).

2.5

-

.

2.5. 107

, 2- (),

R3.

.

2.5.1 . S f : S Rm. f p S (differentiable at p ), - (U, r,W ) S, p W

(2.5.1) F := f r : U Rm

S W f- Rm

R2 U

r

6

F

-

2.2

r1(p). f (differentiable), p S.2.5.2 . f : S Rm p S - .

. (U, r,W ) S, p W F = f r : U Rm r1(p). (U, r,W ) S p W , F := f r r1(p). , Wo := W W S ( W ),

Uo := r1(Wo) U

-

. F Uo

F |Uo = f r|Uo = (f r) (r1 r).

r1(p), r1 r (: ) r1(p) F = f r ( ).

, -

, -

, .

108 2.

2.5.3 . f : S Rm p S, p.

. F = f r f |W = F r1. r F r1(p) ( ), f |W p, .

, , r .

, r1,

2.5.4 . (U, r,W ), r1 : W U R2 .

. W (U, r,W )., ,

r1 r = idU : U U, .

,

, -

.

2.5.5 . A R3 , F : A Rm S S A. f := F |S : S Rm .

. p S. (U, r,W ) S p W , f r = F r : U Rm. F r ( ) f r U , r1(p). f p, S.

2.5.6 . f : S Rm F : A Rm, A R3 S A.

S1, S2 f : S1 S2. - f R3, .

f () -, S2. ,

2.5. 109

, -

,

( 2.5.9

). , ,

.

2.5.7 . S1, S2, f : S1 S2. f p S1 ( 2.5.1, f : S R3) (Ui, ri,Wi) (i = 1, 2) Si , : p W1, f(W1) W2

(2.5.2) F := r12 f r1 : U1 U2 r11 (p).

F (local representation) f (Ui, ri,Wi), i = 1, 2. (2.5.2) .

W1f - W2

R2 U1

r1

6

F- U2

r12

? R2

2.3

. f : S1 S2 p S1 f : S R3. Monge(U2, r2,W2) S2 f(p) W2. f : S1 R3 p, ( ) f : S1 S2 ( S2 ). , W2 S2, A p S1 f(A) W2. (U, r,W ) S1 p W . 2.3.4, (

U1 := r1(W A), r1 := r|U1 , W1 := W A

) S1 p. (Ui, ri,Wi) (i =1, 2) p W1 f(W1) W2, - f

F = r12 f r1 : U1 U2.

110 2.

( Monge) r12 = |W2, : R3 R2 : (x, y, z) (x, y),

F = |W2 f r1 : U1 U2. r11 (p), |W2(. 2.5.5) r11 (p) f r1 ( f p). .

, (2.5.2) r11 (p). r2 ( ),

r2 F = f r1 : U1 R3

r11 (p), . f : S R3 p.2.5.8 . 1) , 2.5.7

. , f : S1 S2 p S1 (Ui, ri,Wi) (i =1, 2) Si , : p W1, f(W1) W2 (2.5.2) r11 (p).

2) 2.5.2, -

f -. , (U i, ri,W i) (i = 1, 2) Si, : p W 1, f(W 1) W 2, (2.5.2) F := r12 f r1 : U1 U2 r11 (p).

3) f : S1 S2 (diffeomorphism) f f1 . , - (U, r,W ), U [. 2.2(1)], 2.5.4 :

r . , R2.

.

.

2.5.9 . Si, i = 1, 2, 3 , f : S1 S2 p S1 g : S2 S3 f(p) S2, g f : S1 S3 p.

2.5. 111

. g f(p), (Ui, ri,Wi) Si, i = 2, 3, f(p) W2, g(W2) W3

G := r13 g r2 : U2 U3 r12 (f(p)). f p ( ), A p S1, f(A) W2. (U, r,W ) S1 p W . - W A S1, p, (

U1 := r1(W A), r1 := r|U1 ,W1 := W A

) S1 (. 2.3.4). p W1 f(W1) W2, f

F := r12 f r1 : U1 U2,

r11 (p) [. 2.5.7 2.5.8(2)]. p W1 (g f)(W1) W3, - (U1, r1,W1) (U3, r3,W3) g f

r13 (g f) r1 = (r13 g r2) (r12 f r1) = G F.

r11 (p), - F r11 (p), G r12 (f(p)) = F (r

11 (p)).

2.5.10 . S , I : I R3 (I) S. to I po :=(to) S. , (U, r,W ) S po, J I to J , : J U R2,

|J = r . 2.10.

. Monge S po ( 2.3.1 ).

(Uo, ro,Wo) ro(u, v) = (u, v, g(u, v)). ,

: R3 R2 : (x, y, z) 7 (x, y)( r1o ) |Wo =r1o . : I R3 ( )

2.6. 113

2.5.11 . , -

= r1|J , J . , , . ( R R3), r1 (. 2.5.1). , -

(

), -

(. 2.5.9).

, -

, .

2.6

S , p S (U, r,W ) 2- S p W . , q := r1(p) U , ru(q) rv(q) (. 2.1.2), 2- R3,

S p (tangent space of S at p) TpS. ,

TpS :={ru(q) + rv(q) | , R

}.

{ru(q), rv(q)} TpS. TpS S p (tangent vectors of S at p).

2.6.1 .

(2.6.1) TpS = [Dr(q)](R2).

. TpS ru(q) + rv(q), (, ) R2, Dr(q)(R2) , ,Dr(q)(a, b), (a, b) R2.

Dr(q)(a, b) = Dr(q)(ae1 + be2) = aDr(q)(e1) + bDr(q)(e2)

= ar

u

q

+ br

v

q

= aru(q) + brv(q),

.

. Dr(q) (a, b)

114 2.

Jacobi :

Dr(q)(a, b) = (Jqr) (a, b)t =

x

u

q

x

v

q

y

u

q

y

v

q

z

u

q

z

v

q

(ab

)

=

(ax

u

q

+ bx

v

q

, ay

u

q

+ by

v

q

, az

u

q

+ bz

v

q

)

= a

(x

u

q

,y

u

q

,z

u

q

)+ b

(x

v

q

,y

v

q

,z

v

q

)

= aru(q) + brv(q).

2.6.2 . Dr(q) : R2 TpS -

. , Dr(q) : R2 R3 11 ( ) TpS = [Dr(q)](R

2).

p S C(S, p)

: J R3, :

0 J, (J) S, (0) = p,

p - S.

Ep :={(0) | C(S, p)}.

2.6.3 .

(2.6.2) TpS = Ep,

TpS (: -), p, p S.

2.6. 115

. w TpS. w = Dr(q)(h), h R2.

: R R2 : t 7 (t) := th+ q.

(0) = q U U R2, > 0 ((, )) U .

:= r (,)

: (, ) R3,

, ((, )) S,(0) = r(q) = p

(0) = (r )(0) = [D(r )(0)](1)= [Dr(q) D(0)](1) = [Dr(q)]((0))= [Dr(q)](h) = w.

TpS Ep., w = (0) Ep, : J S R3 -

(0) = p. 2.5.8,

: (, ) J U R2,

(0) = q |(,) = r .

w = (0) = [D(0)](1) = [D(r )(0)](1)= [Dr((0)) D(0)](1) = [Dr(q)]((0)).

(0) R2, w = [Dr(q)]((0)) TpS, Ep TpS, .

2.6.4 . S p S .

2.6.5 . Ep p S - ru rv S p (tangent plane). TpS p, Ep(S) , , Ep.

116 2.

Ep p S. , p, rU rV .

N

P

vr

ur

p pE T S

2.11

2.7

, ,

. , -

.

.

2.7.1 . S1, S2 f : S1 S2 . : I S1 , f : I S2 .

. f : I S2 t I. to I po = (to). (U, r,W ) S1 po W . : J U , |J = r (. 2.5.8).

f |J = f (r ) = (f r) ,

, f r f (. 2.5.1 2.5.2) .

2.7. 117

2.7.2 . S1, S2 f : S1 S2 - p S1, () f p(differential of f at p)

dpf Tpf : TpS1 Tf(p)S2 : w 7 dpf(w) := (f )(0), : (, ) S1 (0) = p (0) = w.2.7.3 . ,

.

.

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