1

634

2012

2

1. 1.1 .

xy. :

1.1.1 :IR2 I R () . ()R2 .

(t)=(x(t),y(t)) x,y:R , C, ( , ). x,y:R ( ) t. . () (.. x2+y2=1), () y=f(x) (.. y=x2), () (.. r=1) . t. . : (1) x2+y2=1 :

(t)=(x(t),y(t))=(t,t), t[0,2]. t x- . 1(t)=(2t,2t), t[0,] .

(2) 122

2

2

=+by

ax (t)=(at,bt),

t[0,2]. (3) H (t)=(x(t),y(t))=(x0+at,y0+bt), tR (x0,y0) ai+bj t . (4) t x=x(t),y=y(t).

2)(,

2)(

tttt eetyyeetxx ==+==

122 = yx x >0 . (5) f:(a,b)R x (t)=(t,f(t)), t(a,b). (t)=(t,t2), tR y=f(x)=x2 ( ). (6) r=f() : (t)=(x(t),y(t))=(f(t)t, f(t)t) . (7) (t)=(t,|t|) 0 . (8) . (t)=(t3,t2), tR y=f(x)=x23 x=0.

3

(9) . 1 x- (0,0) P (0,0). P x- . P t. t x- (t,0) (t,1) P t. P

32 t x- P=(t,t) P=OK+KP=(t,1)+

+(t,t). (t)=(tt,1t), t0 . t=2 P x-. t=2n n . t=2n .

( ). = ( )t0 0 . ( )t0 .

. 1.1.2. :IR2 ( )t 0 t . t0 (t0) (t0+h)

( ) ( )t h th

0 0+ ( )t0 0

r h0 h0

(t0) ( )t0 0 . (t0)

)()()()( 000 ttsts += , sR. . . .. (t)=(t3,t3), tR y=x t=0. . , 1.1.1 :IR2 a . >0 (a,a+)I ((a,a+)) R2 y=f(x) x=g(y) ( ).

. (t)=(p(t),q(t)) ( )a 0 p a q a( ), ( ) 0. .. >p a( ) 0 . (a,a+)I >p t( ) 0 t(a,a+). p (a,a+) p1:J(a,a+). f(x)=q(p1(x)). f:JR ''(x,y)=(p(t),q(t)) t(a,a+)'' ''y=f(x)'' ( x= p(t)) ((a,a+))R2 y=f(x), xJ. ( q a( ) 0 x=g(y)). . (t)=(x(t),y(t))=(t,t), t[0,2] t= (()=(1,0))

+

2,

2

4

x y= 1 2 , y(1,1) () y=f(x) (0,1) x=1+ . 1.2 . . . :

1.2.1 :[a,b]R2 ((t)=(x(t),y(t))). l() :

l t dt x t y t dta

b

a

b( ) ( ) ( ( )) ( ( )) = = + 2 2 . (t)=(tt,1t), t[0,2] :

===+= 202020 22 .822)1(2)1()( dttdttdtttl

1.2.2 :IR2, :JR2, . , :IJ (s)0 I ( 1 ) :

(s)=((s)) sJ. ,

, . , : l()=I |(s)|ds=I |((s))||(s)|ds=J | (t)|dt=l(), .

. H (s)=(2s,2s), s[0,] (t)=(t,t), t[0,2] : [0,][0,2] (s)=2s. |(t)|=1 t. 1. 1.2.3 :IR2 |(t)|=1 tI.

[,]I t . 1.2.1 :IR2 . :JR2 , .

. . : ( ) ( )t u duat= tI.

(t)=|(t)|0 tI. :I(I)=J =1:JI. :JI (s)=((s)) |(s)|=| ((s))||(s)|=( )=|((s))||((s))|1=1 sJ, .

.

. (t)=(ett,ett), t[0,+). |(t)|2= =[et(t+t)]2+[et(t+t)]2=2e2t. (t)= 2 2 10 eudu ett = ( ) (s)= log( )s

21+ :

))12

log()12

(),12

log()12

(()( ++++= sssss s[0,+). 1.3 .

5

:IR2 . sI () T(s)=(s) . n(s) (T(s), n(s)) R2 (. (i, j)), . T,n s. (s)=(x(s),y(s)) :

T(s)=(x(s),y(s)) n(s)=(y(s), x(s)) sI. 1.3.1 :IR2

|(s)|=1 sI (s).(s)=0 sI sI (s) (s) .

(s).(s)=|(s)|2=.=1. T T(s)= (s)=(x (s),y (s)) n(s) (;) k:IR T(s)=k(s)n(s) sI. 1.3.1 ( ) :IR2 k:IR T(s)=k(s)n(s) sI.

1. + , .

Tn=k ( ) (s) (s).k=T(s)k(s)n(s).k=k(s). k(s)=x (s)y(s)x(s)y(s). ( ) .

1.3.2 :IR2 ( . ) (t)=(x(t),y(t)). :

k t x t y t x t y tx t y t

( ) ( ) ( ) ( ) ( )( ( ) ( ) ) /

= + 2 2 3 2 tI. .

( 1.2.1). s (s)(s).k. (s)=((s)) (s)= ((s))(s) (s)= ((s))((s))2+ ((s)) (s) ((s)) ((s))=0 (s) (s).k=(((s)) ((s)).k)( (s))3=(x((s))y ((s))x((s))y((s))) ((s))3. 1.2.1 t=(s) (s)=|(t)|1 t=(s) k(t)=(x(t)y(t)x( t)y(t)) |(t)|3.

. 1) (t)=(x0+at,y0+bt) 0. 0 =0 s0 (s)=(s0)+(ss0)(s0) . 0 . 2) (t)=(t,t) 1/ . 3) y=f(x) (t,f(t))

: k x f xf x

( ) ( )( ( ) ) /

= + 1 2 3 2 . :IR2 .

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

6

T , :IR . ( (s) 2. 0 s-0I T(s0)=(0,0) :IR T(s)=((s),(s)) (s0)=0). n(s)=((s),(s)) (s)=T(s)= ( (s)(s),(s)(s))

k(s)=(s) sI.

x . :

1.3.1 ( ) I k:IR . :IR2 k. :IR2

. T(s)=((s),(s)) k(s)= (s) sI. (s)=! k(s)ds+C C . T(s)=(x(s),y (s)) (s)=(!(! k(s)ds+C)ds+a, !(! k(s)ds+C)ds+b) a,b . ' ' a,b C. a,b C .

1.3.2 k=k(s) ( ) .

1.3.1 k=k(s) , , . 1 . . .. k=1 (s)=s+C (s)=((s+C)+a,(s+C)+b) ( s) 1. .

k0 I =(s)=k(s)ds d=k(s)ds k , k=k(), :

()=( R()d, R()d ) R()= 1k ( ) .

. k=1/s, s>0, ( ) (s)=logs s=e, R k()=e. ()=( ed, ed)=( 1

2e(+), 1

2e(+)), R.

|R(s)| (s). s0I k(s0)0. s1,s2,s3I s0 C(s1,s2,s3) r(s1,s2,s3) (s1),(s2),(s3). k(s0)0 s1,s2,s3s0 s0 . s0. C(s0) r(s0) . f(s)=|(s)C(s1,s2,s3)| 2 f(s1)=f(s2)=f(s3) s*,s** s1,s2,s3 f (s*)=f(s**)=0. f f(s)=2(s).((s)C(s1,s2,s3)) f(s)=2(s).((s)C(s1,s2,s3))+2|(s)| 2=2k(s)n(s).((s)C(s1,s2,s3))+2

7

(s*).((s*)C(s1,s2,s3))=0 k(s**)n(s**).((s**)C(s1,s2,s3))+1=0. s*,s**s0 s1,s2,s3s0. (s0).((s0)C(s0))=0 k(s0)n(s0).((s0)C(s0))=1.

(s0)C(s0)= 10k s( )

n(s0)=R(s0)n(s0) r(s0)=|(s0)C(s0)|=|R(s0)|. s0

C(s0)=(s0)+R(s0)n(s0) ( ) |R(s0)|. |x(s0)R(s0)n(s0)|=|R(s0)|. : 1 s0 ( ) (s0) 1 . 1|R(s0)|. 1.4 . 1.4.1 :[a,b]R2 : (a)=(b), (a)= (b), (a)=(b),..., (n)(a)=(n)(b),... .

. :RR2 ( ba). (t)=(t,t), t[0,2] (t)=(2tt, 2tt), t[0,/2] (0)=(/2).

1.4.2 :[a,b]R2 (t1)(t2) t1,t2[a,b) t1t2.

:[0,l]R2 (s)=((s),(s)) (s)= k(s)ds+C. (0)= ( l) (l)(0)=2m m . m T(s)=(s) .

1.4.3 m= 12 0 k s ds

l ( ) .

( ) .

1.4.1 (Jordan) :[0,l]R2 R2\([0,l]) ( ) ([0,l]).

1.4.2 ( ) :[0,l]R2 1.

=l dssk0 2)( . 1.4.3 ( ) l

A R2\([0,l]) l24A . . 1.4.3 :[a,b]R2 t[a,b] ([a,b]) () (t).

( ): 1.4.3

k ,

8

R2.

, . :

1.4.4 :[a,b]R2. (t0), t0[a,b] t0 k (t0)=0.

122

2

2

=+by

ax (a0} y(s)>0 + y(s)

9

9. :IR2 k(t)0 I ( ) ( )( )

t tk t

= + 1 n(t) . (i) k (t)0 (t) (t). (ii) (t)=(at,bt), t[0,2].

10. () :[0,]R2 : (i) k(s)=2+2s, (ii) k(s)=1+8s, (iii) k(s)=4(s+16s);

11. * f(x,y)=0 ( f(x,y)0 (x,y) f(x,y)=0).

12. :[0,l]R2 0

10

2) (t)=(x(t),y(t),z(t))=(at,at,bt), tR xy- z-. s=(t)=

( )u dut0 =t 22 ba + ' .

:IR3 . sI () T(s)=(s) . n :

2.1.1 ( ) :I->R3 k:IR k(s)=|T (s)|=| (s) | sI.

k(s)0 sI. ( 1). (s) . (s)=0 (s)0 n(s) T(s)

(s)=T(s)=k(s)n(s). n(s) ( ) . 2.1.1 (s) T(s)

n(s) s.

. s0I k(s0)0. s1,s2,s3I s0 (s1),(s2),(s3). a(s1,s2,s3).x=c(s1,s2,s3) a(s1,s2,s3) . k(s0)0 s1,s2,s3s0 s0 . a(s0).x=c(s0) . f(s)=a(s1,s2,s3).(s) f(s1)=f(s2)=f(s3) s*, s** s1,s2,s3 f(s*)=f (s**)=0. s*,s**s0 s1,s2,s3s0 a(s0).(s0)=a(s0).(s0)=0 k(s0)0 a(s0).T(s0)=a(s0).n(s0)=0. (s0). :

(s0)(s0).(x(s0))=0 :

0)()()()()()(

)()()(

000

000

000

=

szsysxszsysx

szzsyysxx.

2.1.3 b(s)=T(s)n(s), (s) (s). (s)+R(s)n(s) R(s)=1k(s) ( ) s. b . k(s)=|T (s)T(s)|=|(s)(s) | :

2.1.1 :IR3 ( . ), (t)=(x(t),y(t),z(t)). :

11

k tt t

t

x t y t z tx t y t z t

x t y t z t( )

( ) ( )

( )

( ) ( ) ( )( ) ( ) ( )

( ( ) ( ) ( ) ) /=

=

+ +

3 2 2 2 3 2

i j k

tI.

1.3.2. 2.2 . Frene't.

:IR3 k(s)0 sI b(s) . b (s) . |b (s)| . b .b=0 ( b ) b=(Tn)=T n+Tn= Tn T=kn b .T=0. b n. 2.2.1 :IR3 k(s)0 sI :IR

b (s)=(s)n(s) sI. sI s. Frene't . . T(s)=k(s)n(s) b (s)=(s)n(s). n (s) n=bT n (s)= b (s)T(s)+b(s)T(s)=(s)n(s)T(s)+k(s)b(s)n(s). :

dds

k

dds

k

dds

T n

n T b

b n

=

= +

=

. . 2.2.2 (i) (s) T(s), n(s) s. b(s).(x(s))=0. (ii) (s) n(s), b(s) s. T(s).(x(s))=0.

(iii) (s) b(s), T(s) s. n(s).(x(s))=0.

=T, =kn kb= k b+kb=+ . k2=kn.(k b+kb)= .()=() . .

( )( )

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

sk s

x s y s z sx s y s z sx s y s z s

=

12 .

:

2.2.1 :IR3 ( . ), (t)=(x(t),y(t),z(t)). :

12

( )( ( ) ( )) ( )

( ) ( )

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( )t t t t

t t

x t y t z tx t y t z tx t y t z t

t t=

=

2 2 . .

. s k(s)2((s)(s)) . (s) k(s)=| (s)(s) | . (s)=((s)) (s)= ((s))(s), (s)= ((s))((s))2+((s))(s) (s)= ((s))((s))3+ +3((s)) (s)(s)+((s)) (s). ((s)(s)) .(s)= =( ((s))(s) ((s))((s))2). ((s))((s))3=(((s)) ((s))). (s))((s))6. |(s) (s) |= | ((s)) ((s))|((s))3. ((s))=|(s)(s) | 2(((s)) ((s))). (s))( (s))6= =|((s)) ((s))|2(((s)) ((s))). (s)) t=(s) .

0 . :

2.2.2 :IR3 k(s)0 sI. (s)=0 sI.

. ( - ). a.x=b a.(s)=b sI a. (s)=a.(s)=a.(s)=0 sI (s),(s),(s) . ((s)(s)).(s)=0 (s)=0 sI. (s)=0 sI b(s) b(s)=a sI. a.(s)=b(s).T(s)=0 a.(s)=b=. sI a.x=b.

. (t)=f(t)a1+g(t)a2+a3 a1,a2,a3 ( k0) .

, , . :

2.2.1 ( ) I k,:IR k(s)>0 sI. :IR3 k . :IR3 .

. s0I . T,n,b:IR3

dds

k

dds

k

dds

T n

n T b

b n

=

= +

=

T(s0)=i,n(s0)=j,b(s0)=k. sI R3. s=s0 (T.n,n.b,T.b,T.T,n.n,b.b) :

13

dds

k k

dds

k

dds

k

T n n n T T T b

n b T b n n b b

T b n b T n

. . . .

. . . .

. . .

= +

= +

=

dds

k

dds

k

dds

T T T n

n n T n n b

b b n b

. .

. . .

. .

=

= +

=

2

2 2

2

I

(T.n,n.b,T.b,T.T,n.n,b.b)(s0)=(0,0,0,1,1,1) (.. (n.b)=n.b+n.b=(kT+b).b+n.(n)). (0,0,0,1,1,1) (T.n,n.b,T.b,T.T,n.n,b.b)=(0,0,0,1,1,1) I sI R3. :

( ) ( )s s ds= T . =T |T(s)|=|k(s)n(s)|=k(s) k(s)>0 n . =(kn) =kn+kn=knk2T+kb k2().=k2(Tkn).(knk2T+kb)= T,n,b . :IR3 A A(s0)=(s0) A, s0 . A, I. A(s0)=(s0) A . .

2.2.3 k=k(s),=(s) ( ) .

1.3.1 k=k(s),=(s)

, , . . .

k aa b

ba b

= + = +2 2 2 2, . .

:IR3 k(s)0 sI 0I ( ) (0)=(0,0,0), T(0)=i,n(0)=j,b(0)=k, R3. Taylor :

(s)=(0)+(0)s+ (0) s2

2+ (0) s

3

6+(s4)

s 0 (s4) r(s) |r(s)|Cs4 s 0. (0)=T(0)=i, (0)=k(0)n(0)=k(0)j (0)=k (0)n(0)+k(0)n(0)=k (0)n(0)+k(0)(k(0)T(0)+(0)b(0))=k(0)2i+k(0)j+(0)k. k=k(0),=(0),k=k(0)

(s)=(sk2 s3

6,k s

2

2+k s

3

6,k s

3

6)+(s4)

s s :

14

=)(s (s, k s2

2, k s

3

6).

0 k0. ( xy-) xy(s)=(s, k

s2

2,0)

y= 12

kx2, ( yz-)

yz(s)=(0, ks2

2, k s

3

6) z2= 2

9

2k

x3

( zx-) zx(s)=(s, 0, ks3

6)

z= 16kx3.

( T) '' '' b . 0 '''' 0 ( x3) ''''. .

. :IR3 .

s0I k(s0)(s0)0. s1,s2,s3,s4I s0 C(s1,s2,s3,s4) r(s1,s2,s3,s4) (s1),(s2),(s3),(s4). k(s0)(s0)0 s1,s2,s3,s4s0 . s0. C(s0) r(s0) . f(s)=|(s)C(s1,s2,s3,s4)|2 f(s1)=f(s2)=f(s3)=f(s4) s*,s**,s*** s1,s2,s3,s4 f(s*)=f(s**)=f(s***)=0. f f (s)=2 (s).((s)C(s1,s2,s3,s4)), f(s)=2(s).((s)C(s1,s2,s3,s4))+2|(s)|2=2k(s)n(s).((s)C(s1,s2,s3,s4))+2 f(s)=2(s).((s)C(s1,s2,s3,s4))+4(s).(s)=2(s).((s)C(s1,s2,s3,s4)). =(kn)=kn+kn=knk2T+kb. s*,s**,s***s0 s1,s2,s3,s4s0

(s0).((s0)C(s0))=0, k(s0)n(s0).((s0)C(s0))=1 (k(s0)n(s0)k2(s0)T(s0)+k(s0)(s0)b(s0)).((s0)C(s0))=0.

R sk s

( )( )

= 1

(s0).((s0)C(s0))=0, n(s0).((s0)C(s0))=R(s0) b(s0).((s0)C(s0))= R ss( )

( )0

0 .

C(s0)=(s0)+R(s0)n(s0)+ R ss( )

( )0

0 b(s0)

r s R s R ss

( ) ( ) ( ( )( )

)0 02 0

0

2= + .

15

2.2.3 :IR3 k(s).k (s).(s)0 sI :

R s R ss

c( ) ( ( )( )

)2 2+ = = . sI. . ' .

R s R ss

c( ) ( ( )( )

)2 2+ = = . sI. 2RR +22R R 23(R )2 =0 1R (R+(R ) )=0 ,R 0 I R+(R ) =0 I. (s)=(s)+R(s)n(s)+(s)1R (s)b(s). =T+R n+R(kT+b)+1R (n)+(R ) b=(R+(R ))b=0 I. (s)=a= |(s)a|= R s R s

sc( ) ( ( )

( ))2 2+ = = . sI

a c. k (s)0 . a2+b2=1, ab0 (s)=(as,as,bs) k=.0, =.0 R2+(R)2 .

1. : (i) (t)=(a(tt),a(1t),bt), tR, (ii) (t)=(3tt3,3t2,3t+t3).

2. (t)=(2t,tt,t), t[0,2]. .

3. :IR3 . .

4. 0, (s)=1n(s)+!b(s)ds ||.

5. :IR3 '''' () . (s)0 sI k(s)(s) . (t)=(6t,3t2,t3), .

6. : k(s)=(s)= 22s

, s>0. 7. b=b(s) (

) 0, .

8. :IR3 ( k0) ( (s) n(s)).

9. :IR3 ( k0) ( (s) b(s)).

10. * :RR3 :R->R (s)= s

2T(s)+(s)n(s)+ s

2b(s), sR. (0)=( 2

2,0,0)

. 11. k(t0)0 (t0)

(t0).

16

12. .

13. (t)=(t,2t,t), tR ( ). t=4 : (i) , (ii) ( ) *(iii) .

14. ,:IR3 sI , s . .

15. * (. ). .

16. :IR3 ( k(s)0 sI). (s)+b(s) sI. 1 .

17. :IR3 k0. (s)=(s)+ 1

kn(s)

k. 18. :IR3

k(s)>0 (s)>0. (s)=b(s).

3. 3.1 . . X:UR3 UR2 X(u,)=(x(u,),y(u,),z(u,)) U. . Xu X

X Xu

xu

yu

zuu

= =

( , , ) X

X x y z

= =( , , ) .

:

xux

zuz

uxz

zuz

yuy

uzy

yuy

xux

uyx ===

),(),(,

),(),(,

),(),( .

. X:R2R3 X(u,)=(x0+a1u+b1,y0+a2u+b2,z0+a3u+b3)

X0=(x0,y0,z0) a=(a1,a2,a3) b=(b1,b2,b3) a,b . a b X(R2) .

17

: 3.1.1 X:UR3 UR2 , q=(u0,0)U Xu(q) X(q) R3 . Xu(q)X(q)0 qU. Xu(q)X(q)0

( , )( , )

( ), ( , )( , )

( ), ( , )( , )

( )x yu

q y zu

q z xu

q .

X:UR3 q=(u0,0)U Taylor :

X(u,)=X(u0,0)+(uu0)Xu(u0,0)+(0)X(u0,0)+R(u,) |R(u,)|C(|uu0| 2+|0| 2) (u,) (u0,0).

q(u,)=X(u0,0)+(uu0)Xu(u0,0)+(0)X(u0,0) Xu(u0,0) X(u0, 0) |X(u,)q(u,)|C(|uu0| 2+|0| 2) (u,) q=(u0,0) q X (u0,0).

N( )( ) ( )( ) ( )

qX q X qX q X q

u

u=

X q. :

N(q).(xX(q))=0. . X(u,)=(u,,u2+2) q=(1,2) Xu(q)=(1,0,2), X(q)=(0,1,4) N(q)=

121

(2,4,1). q(u,)=(u,,2u+45) 2x4y+z=3. w R3 X q . ,R w=Xu(q)+X(q). R3 Xu(q) X(q) . TqX X q. (DX)q:R2R3 ( 3.1.1 ) X q (DX)q((1,0))=Xu(q) (DX)q((0,1))=X(q)

TqX=(DX)q(R2) X q X q. u=. =. X . (u)=X(u,c1) ( =c1 ) Xu ()=X(c2,) ( u=c2 ) X. : 3.1.2 Y:VR3 X:UR3 (C) :VU ()

Y(s,t)=X((s,t)) (s,t)V.

18

X,Y . (s,t)=(u(s,t),(s,t)) :

Y X us

Xs

Y X ut

Xts u t u

= + = +

,

Y Y

us

ut

s t

X X us t

X Xs t u u = =

( )( , )( , )

( ) 0

. Y X,Y ,

TbYT(b)X bV. X,Y . . X(u,)=(u,,u) Y(s,t)=(s+t,st,s2t2) u=s+t,=st ( (s,t)=(s+t,st)) R2. ( X ) X pX(U) ( p U). , , : 3.1.3 X:UR3 X:UX(U) U R2 X(U) R3 . S=X(U) R3 X. 3.1.1 X:UR3 X1:X(U)U . VU X(V) X(U), WR3 X(V)=X(U)W. . X . . . 1) . UR2 f:UR '''' X:UR3 X(u,)=(u,,f(u,)) . Xu=(1,0,fu), X=(1,0,f) X X1:X(U)U pr:R3R2 pr(x,y,z)=(x,y) X(U). 2) X:U(=(0,2)(0,))R3 X(u,)=(u,u,) . S2={(x,y,z)R3:x2+y2+z2=1} . N(u,)=X(u,) . u, . u=. =. . S2 S2 R2. ( 1.1.1): 3.1.1 X:UR3 qU. VU q X|V:VR3

19

( X V) ( ) z=f(x,y), x=g(y,z), y=h(z,x). X:UR3 pX(U) WR3 pW X(U)W .

. ( , )( , )

( )x yu

q , ( , )( , )

( )y zu

q ,

( , )( , )

( )z xu

q .

( , )( , )

( )x yu

q 0. pr:R3R2 pr(x,y,z)=(x,y) :UR2 (u,)=proX(u,)=(x(u,),y(u,)). q

( , )( , )

( )x yu

q 0. VU q :V(V) V A=(V)R2. Y:AR3

Y(s,t)=X(1(s,t)) (s,t)A. H X|V , . Y . (s,t)A

proY(s,t)= proX(1(s,t))=(1(s,t))=(s,t) Y(s,t)=(s,t,f(s,t)) f:AR . .

( , )( , )

( )y zu

q 0 x=g(y,z) ( Y(s,t)=(g(s,t),s,t))

( , )( , )

( )z xu

q 0 y=h(z,x) ( Y(s,t)=(t,h(s,t),s)). X X(V) X(U), WR3 X(V)=X(U)W. S2 , , : 3.1.3 S R3 pS WR3 p X:UR3 X(U)=SW. X:UR3 S p. X1:SWU . S {Xi:UiR3, i} ( I ) {Xi(Ui), i} S. ( ). . S2={(x,y,z)R3:x2+y2+z2=1} . U={(u,)R2: u2+2

20

. . WR3 F:WR aF(W) F pW F(p)=a F(p)0 ( Fx(p),Fy(p),Fz(p) ). 3.1.2 WR3 , F:WR aF(W) F.

S=F1({a})={(x,y,z)W : F(x,y,z)=a} . pS=F1({a}) F(p)(0) S p. . p=(x0 ,y0 ,z0)S. Fz(p)0. - W1W p UR2 (x0 ,y0) g:UR g(x0 ,y0)= z0 ''(x,y,z)W1 F(x,y,z)=a'' ''(x,y)U z=g(x,y)''. X:UR3 X(u,)=(u,,g(u,)) () X(U)=SW1.

Xu(x0,y0)X(x0,y0)=(1,0,gx(x0,y0))(0,1,gy(x0,y0))=(gx(x0,y0),gy(x0,y0),1) S p. F(x,y,g(x,y))=a

Fx(x,y,g(x,y))+Fz(x,y,g(x,y))gx(x,y)=0 Fy(x,y,g(x,y))+Fz(x,y,g(x,y))gy(x,y)=0. F(x0 ,y0 ,z0) Xu(x0,y0)X(x0,y0):

F(x0 ,y0 ,z0)=Fz(x0 ,y0 ,z0)Xu(x0,y0)X(x0,y0)0. Fx(p)0 Fy(p)0.

. 1) W=R3\{0} a>0 F:WR F x y z x y a z x y z a a x y( , , ) ( )= + + = + + + +2 2 2 2 2 2 2 2 2 22 .

0

21

f x y x y( , )= +2 2 (0,0). K . . 3.1.3 ( ) S , pS X:US Y:VS S p, pX(U) Y(V)=A.

h=X1oY : Y1(A)X1(A) R2. X:US Y:VS p. . A S X,Y Y1(A),X1(A) R2 h=X1oY . qY1(A) Z S Y(q)A ( 3.1.1). z=f(x,y) Z(s,t)=(s,t,f(s,t)) q ( -). h=(Z1X)1(Z1Y) q Z1X Z1Y q. X(u,)=(x(u,),y(u,),z(u,)) , z(u,)=f(x(u,),y(u,)) q

Z1X(u,)=Z1(x(u,),y(u,),f(x(u,),y(u,)))=(x(u,),y(u,)) q

( , )( , )

( )x yu

q . 0

(DX)q ( q1=(x(q),y(q)))

xu

q x q

yu

q y q

fx

q xu

q fy

q yu

q fx

q x q fy

q y q

( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 1+ +

2,

, . ( , )( , )

( )x yu

q 0 Z1X q. Z1Y q. 3.1.1 X:UR3 :IR3 (I)X(U) ( ). =X1:IR2 . . X:UX(U) . t0I Z '''' (t0) X(U) ( 3.1.1). t0 (t)=(X1Z)(Z1)(t) 3.1.3 X1Z Z1 t0. Z1 . .. Z(u,)=(u,,f(u,)) p=(x,y,z)X(U) (t0) Z1(p)=(x,y). (t)=X(u(t),(t)). . . S2 ( ) . N=(0,0,1) '' '' S=(0,0,1) ''''. xy.

22

'' '' N :S2\{N}R2 S :S2\{S}R2

N Sx y z x zy

zx y z x

zy

z( , , ) ( , ), ( , , ) ( , ).= = + +1 1 1 1

X=N1:R2S2\{N} Y=S1:R2S2\{S} .

)11,

12,

12(),( 22

22

2222

++++

++++= uu

uuuuX (u,)R2

)11,

12,

12(),( 22

22

2222 tsts

tst

tsstsY ++

++++= (s,t)R

2.

X(R2) Y(R2)=S2\{N,S}, X1(S2\{N,S})=Y1(S2\{N,S})=R2\{(0,0)} h=X1oY : R2\{(0,0)}R2\{(0,0)} :

h s t ss t

ts t

( ,. ) ( , )= + +2 2 2 2 (s,t)R2\{(0,0)}.

: 3.1.4 S pS S p TpS ( 2) TqX R3 X:UR3 S p X(q)=p.

TpS=(DX)q(R2). TpS . TpS . 3.1.1 TpS w R3 >0 :(,)SR3

(0)=p (0)=w. . X:US S p. wTpS ,R w=Xu(q)+X(q) q=X1(p). (t)=X(q+(,)t), |t|0 q+(,)tU |t|

23

Y11ofoX1=(Y11oY)o(Y1ofoX)o(X1oX1) C X11(p) (X1oX1), (Y11oY) . . f:S2S2 f(x,y,z)=(x,y,z) . .. X u u u( , ) ( , , ) = 1 2 2 ( ) Y Y u u u( , ) ( , , ) = 1 2 2 Y1ofoX(u,)= =(u,) . : 3.2.1 F:WR3 C WR3 S1,S2 S1W F(S1)S2. f= F S1 :S1S2 . F:WR C f= F S1 :S1R . . f pS1 () X:US1,Y:VS2 S1,S2 p,f(p) f(X(U))Y(V) Y:VS2 . Y1 FX:UR3 C . F(S1) S2 Y1ofoX= Y1oFoX U C. . . p0,uR3 S f,h:SR f(p)=|pp0|2 h(p)=u.(pp0) C R3. 3.2.2 f:S1S2 f:S1S2 . S1,S2 .

. S2 S={(x,y,z)R3: 122

2

2

2

2

=++cz

by

ax }

. f:S2S f(x,y,z)=(ax,by,cz) ( 3.2.1) f 1(x,y,z)= =(xa,yb,zc) .

'' '' f:S1S2 pS1 '''' f p.

3.2.3 f:S1S2 pS1. f p, (df)p:TpS1Tf(p)S2 : wTpS1 :(,)S1 (0)=p (0)=w (df)p(w)=(0) =fo:(,)S2.

3.2.2 (df)p:TpS1Tf(p)S2 . . X:US1,Y:VS2 () S1,S2

p,f(p) f(X(U))Y(V) h=(h1,h2)=Y1ofoX:UV f. w=Xu(q)+X(q)TpS1, q=X1(p) :(,)S1 (0)=p (0)=w. t (t)=X(u(t),(t)) w=(0)=u(0)Xu(q)+(0)X(q) ' =u(0),=(0). (t)=fo(t)=foX(u(t),(t))=Yoh(u(t),(t))=Y(h1(u(t),(t)),h2(u(t),(t)))

(df)p(w)=(0)=Ys(h(q)) ( ( ) ( ) )

hu

q h q1 1+ +Yt(h(q)) ( ( ) ( ) )

hu

q h q2 2+ . (df)p(w) , (,).

24

. (df)p:TpS1Tf(p)S2

)()(

)()(

22

11

qh

quh

qh

quh

f. f:SR

pS ( ) (df)p:TpSR (df)p(w)=(fo)(0) wTpS :(,)S (0)=p (0)=w. (.. ) . :

3.2.3 f:S1S2 pS1. f p, (df)p:TpS1Tf(p)S2 f .

. 1) L:R3R3 S1,S2 L(S1)S2. 3.2.1 L:S1S2 . wTpS1 :(,)S1 (0)=p (0)=w. (dL)p(w)= lim ( ( ( )) ( ( )) ) lim ( ( ) ( ) ) ( ( ))t tL t Lt L

tt

L = = 0 00 0 0 =L(w)

(dL)p L TpS1 R3. 2) f:SR f(p)=|pp0|2. wTpS

:(,)S (0)=p (0)=w (df)p(w)=(fo)(0)= ddt t=0

|(t)p0|2=2((0) p0).(0)=2(p p0).w. :

3.2.4 F:WR3 WR3 S1,S2 S1W, F(S1)S2 f= F S1 :S1S2. (df)p:TpS1Tf(p)S2 f pS1 (DF)p:R3R3 TpS1 R3. . wTpS1 :(,)S1 (0)=p (0)=w (df)p(w)=(fo)(0)=(Fo)(0)=(DF)(0)((0))= =(DF)p(w). 3.3 Gauss '' ''. S . N:SS2R3 pS N(p) TpS . , .. Mobius N. ( ) N . . . 3.3.1 N:SS2R3 pS N(p) TpS N Gauss S. S N.

25

. 1) X:UR3 S=X(U) S ))(()( 1 pX

XXXX

pu

u =

N .

N(u,) N(X(u,)), (u,)U. 2) S F(x,y,z)=a

3.1.2 N( ) ( )p FF

p= pS.

Gauss N pS S p. N ( TpS) ( ) (dN)p:TpSTN(p)S2.

TN(p)S2={wR3:w.N(p)=0}=TpS, (dN)p:TpSTpS.

3.3.2 (dN)p:TpSTpS S p.

: wTpS :(,)SR3 (0)=p (0)=w, N(t)=No(t). N(t) (dN)p(w)=N(0) N(0)=N(p) TpS. X:US S p w=Xu(q)+X(q)TpS, q=X1(p) :(,)S (0)=p (0)=w (t)=X(u(t),(t)) =u(0),=(0) (dN)p(w)=Nu(q)+N(q) ( N(u,) N(X(u,)).

(dN)p(Xu(q)+X(q))=Nu(q)+N(q) ,R. S F(x,y,z)=a 3.1.2 3.2.4

(dN)p:TpSTpS N( ) ( )p FF

p= ( R3)

S. . 1) N( )p p= 3.2.4 (dN)p=id T Sp 2 . () . 2) S x2+y2=1 p=(1,0,0) N(x,y,z)=(x,y,0) (x,y,z)S 3.2.4 (dN)p(,,)=(,,0) (,,)TpS={(,,)R3:=0}. (dN)p(j)=j (dN)p(k)=0 p y- z-.

1. P={(x,y,z)R3:x=y}. X:UP U={(u,)R2:u>}

X(u,)=(u+,u+,u) () P. 2. ; (i) x4+y6+z2=1, (ii)

(x+y)3+y3+(zx)3=2, (iii) x4=(y+z)4, (iv) exy+eyz+ezx=4, (v) x2+xy+y2=z3. 3. l1,l2 l1 z l2 y=1,z=0. S

l1 l2 l1 l2 2. (i) S . *(ii) S{(0,0, 3 )} .

26

4. z xa

yb

= +2

2

2

2 z=0

. 5. N=(0,0,1) S2, N :S2\{N}R2

X=N1:R2S2\{N}. h:R2R2 h(x,y)=(ax2+by2,xy) a,bR. a,b f:S2S2 f(p)= X h X p p S N

N p No o

=

1 2( ) \{ }

N. 6. p=(x0,y0,z0)

: (i) x4+y4+z4=1, (ii) x2+2y2z2=2, (iii) z=(xy). 7. S P SP={p} P

S p. 8. X:R2R3 X(u,)=(u,u+,2u2+2)

q=(1,1)R2. V q ( ) X|V:VR3.

9. S={(x,y,z)R3:x2y+y2z+z2x=6} p=(2,2,1) .

10. S f:SR (df)p=0 pS, f .

11. S pS. (i) f:SR p f (df)p=0. (ii) f(x,y,z)=xyz S2.

12. f:S2S S2 S= ={(x,y,z)R3:4x2+2y2+z2=1} f(p)='' S 0p'' pS2. f (df)p:TpS2Tf(p)S pS2.

13. S S .

14. S=X(R2) X(u,)=( u, u,u). (i) S . (ii) N(u,) S. (iii) p=(1,0,), w=(3,2,2) wTpS w Xu(q), X(q) q=X1(p).

3.4 . R3 . ' . p S < , >p:TpSTpSR R3 TpS. p, R3 , . 3.4.1 Ip:TpSR Ip(w)=p=w.w0 wTpS S pS. X:US () S p. w=Xu(q)+X(q)TpS, q=X1(p)

Ip(w)=(Xu(q)+X(q)).(Xu(q)+X(q))=2Xu(q).Xu(q)+2Xu(q).X(q)+2X(q).X(q). E,F,G:UR

E(u,)=Xu(u,).Xu(u,), F(u,)=Xu(u,).X(u,), G(u,)=X(u,).X(u,), (u,)U,

Ip(Xu(q)+X(q))=2E(q)+2F(q)+2G(q) qU, p=X(q)S, ,R. w1=1Xu(q)+1X(q),w2=2Xu(q)+2X(q)TpS,

p=12E(q)+(12+21)F(q)+12G(q).

27

E,F,G:UR X(U) S X:US. S. :IS (t)=X(u(t),(t)) (t)=Xu(u(t),(t))u(t)+X(u(t),(t))(t) atb :

dttttuGttuttuFtuttuEdttl ba

b

a p ++== 22 )())(),(()()())(),((2)())(),(())(()( w1,w2TpS

=< >w w

w w1 2

1 2

,

( ) ( )p

p p

' S. u=u0 =0 X:US p=X(u0,0)

=),(),(

),()()(

,

0000

00

uGuE

uFXX

XX

pup

pu =>0 (a,b). S C z X(u,)=(()u,()u,()), u(0,2), (a,b) Y(u,)=(()u,()u,()), u(,), (a,b) . S . u=. =. . X S\C Xu(u,)=(()u,()u,0), X(u,)=(()u,()u,()) (u,)=

22 )()(1

+ (()u,()u,()). y=x S z=(0) S (0)=0 C 0 z.

E(u,)=()2, F(u,)=0, G(u,)=()2+()2, X . S ( )

+=+= baba ddudSA 2220 22 )()()(2)()()()( . C = ba dSA )(2)( =2(ba).( C z). 3) . S h:UR, UR2 . X:UR3 X(x,y)=(x,y,h(x,y)) Xx(x,y)=(1,0,hx(x,y)), Xy(x,y)=(0,1,hy(x,y))

N= 1

1 2 2+ +h hx y(hx,hy,1).

E h F h h G hx x y y= + = = +1 12 2, , .

x=. y=.

=22 11 yx

yx

hh

hh

++

. EG F h h h h h hx y x y x y = + + = + +2 2 2 2 2 21 1 1( )( ) ( ) , S RU

A(X1(R))= ++R yx dxdyhh 221 . '' '' . E,F,G:UR () S X:US X S .

29

3.4.2 f:S1S2 pS1 w1,w2TpS1

< > =< >w w w w1 2 1 2, ( ) ( ),( ) ( ) ( )p p p f pdf df . US1 , f:US2 . . pS1 wTpS1 Ip(w)=If(p)((df)p(w)) ( 2= ). . A:R3R3 A A:S2S2 . 3.4.1 S,S* , UR2 X:US, X*:US* E=E*, F=F*, G=G* U f=X*oX1:X(U)S* . . f X*1ofoX=idU. pS q=X1(p)=X*1(f(p)) (df)p idU q (df)p(Xu(q)+X(q))=Xu*(q)+X*(q) ,R. Ip(Xu(q)+X(q))=2E(q)+2F(q)+2G(q) If(p)(Xu*(q)+X*(q))=2E*(q)+2F*(q)+2G*(q) E=E*, F=F*, G=G* U Ip(w)=If(p)((df)p(w)) pS. f . . S xy- S* x2+y2=1 X:US, X*:US* U=(0,2)R X(u,)=(u,,0), X(u,)= =(u,u,) E=E*=G=G*=1, F=F*=0, U X*oX1:X(U)S* .

1. : (i)

X(u,)=(au,bu,c) (), (ii) X(u,)=(au,bu,2) (- ), (iii) X(u,)=(uu,u+u,). - (i) .

2. (i) S2

( ) X u uu u

uu

( , ) ( , , )

= + + + +

+ ++ +

21

21

112 2 2 2

2 2

2 2 , (u,)R2. (ii) S2 u(t)=2+t, (t)=3t, tR. (iii) A={(u,)R2:u>0, 0

30

7. z=x2+y2. (i) za2. (ii) x=. .

8. X(u,)=(u,u,log+u), < 0 x2+y2z2=0} xy-.

12. * X:US2 E(u,)=G(u,)=1, F(u,)=0 (u,)U. .

3.5

(dN)p:TpSTpS S p Ip :

3.5.1 (dN)p:TpSTpS S p () p TpS.

. p=p w1,w2TpS. X:US S p. (dN)p(Xu(q)+X(q))=Nu(q)+N(q) ,R q=X1(p)

Nu(q).X(q)=N(q).Xu(q). N.X=0 u Nu.X+N.X u=0 N.Xu=0 N.Xu+N.Xu=0. X u=Xu U Nu.X=N.X u=N.Xu=N.Xu U (dN)p . 3.5.1 IIp:TpSR ( (dN)p) IIp(w)=p wTpS S pS. . X:US S p ,R :

IIp(Xu(q)+X(q))=(Nu(q)+N(q)).(Xu(q)+X(q)) IIp(Xu(q)+X(q))=2N(q).Xu u(q)+2N(q).Xu(q)+2N(q).X(q).

e,f,g:UR e(u,)=N(u,).Xu u(u,), f(u,)=N(u,).Xu(u,), g(u,)=N(u,).X(u,)

S X:US :

IIp(Xu(q)+X(q))=2e(q)+2f(q)+2g(q) qU, p=X(q)S, ,R. . ( N( )p p= ) (dN)p=id T Sp 2

IIp=Ip pS2.

:

31

) (dN)p:TpSTpS . TpS k1(p),k2(p) (k1(p)k2(p)) ( (dN)p) (dN)p(w1(p))=k1(p)w1(p) (dN)p(w2(p))=k2(p)w2(p). ) k1(p) k2(p) IIp {wTpS : Ip(w)=1} TpS. 3.5.2 ) w1(p),w2(p) S p. (dN)p . ) k1(p), k2(p) (k1(p)k2(p)) ( (dN)p) S p. ) K(p)=k1(p)k2(p)=det((dN)p) - Gauss S p.

) H(p)= 12

(k1(p)+k2(p))= 12

tr((dN)p) -

S p. X:US S p. (dN)p(Xu(q)+X(q))=Nu(q)+N(q) ,R ( q=X1(p)) (dN)p Xu(q),X(q) TpS :

Nu ua X a X= +11 21 , N = +a X a Xu12 22 .

(dN)p(Xu(q)+X(q))=(a11(q)Xu(q)+a21(q)X(q))+(a12(q)Xu(q)+a22(q)X(q)) ,R

(dN)p(Xu(q)+X(q))=(a11(q)+a12(q))Xu(q)+(a21(q)+a22(q))X(q) ,R (dN)p Xu(q),X(q) TpS

a q a qa q a q

11 12

21 22

( ) ( )( ) ( )

.

Xu(q),X(q) TpS . aij:UR Xu X :

= = + = + = = + = + = = + = + = = + = +

e X a X X a X X a E a Ff X a X X a X X a F a Gf X a X X a X X a E a Fg X a X X a X X a F a G

u u u u u

u u

u u u u

u

NNNN

11 21 11 21

11 21 11 21

12 22 12 22

12 22 12 22

,,,,

=

e ff g

E FF G

a aa a

. 11 1221 22

' a aa a

E FF G

e ff g EG F

G FF E

e ff g

11 12

21 22

1

21

=

=

. . .

32

N

N

u u u

u u

fF eGEG F

X eF fEEG F

X

gF fGEG F

X fF gEEG F

X

= +

= +

2 2

2 2

,

,

Weingarten. Gauss:

K=det(ai j)=eg fEG F

2

2 ,

:

H= 12

tr(ai j)= 12

(a1 1+a2 2 )=eG fF gE

EG F +

2

2 2( ).

k1, k2 (k1k2) k22Hk+K=0 : k H H K k H H K1

22

2= + = , . k1, k2 (k1k2) S k1>k2 H2>K. S k1(p)=k2(p) . (dN)p=kid T Sp ( k=k1(p)=k2(p)) S p () .

(dN)p k1(p), k2(p). w=Xu(q)+X(q) (dN)p

k(Xu(q)+X(q))=(dN)p(Xu(q)+X(q))=Nu(q)+N(q) Xu(q) X(q) :

e(q)+f(q)=k(E(q)+F(q)) f(q)+g(q)=k(F(q)+G(q)) k (E(q)+F(q))(f(q)+g(q))=(F(q)+G(q))(e(q)+f(q)) 2(E(q)f(q)F(q)e(q))+(E(q)g(q)G(q)e(q))+2(F(q)g(q)G(q)f(q))=0

2 20

=E q F q G q

e q f q g q( ) ( ) ( )( ) ( ) ( )

.

. Ip(w)=2E(q)+2F(q)+2G(q)=1.

p S ,R R

e(q)=E(q), f(q)=F(q), g(q)=G(q). : 3.5.1 p=X(q) S . Xu(q) X(q) f(q)=F(q)=0.

e qE q

( )( )

g qG q

( )( )

.

. Xu(q) X(q) F(q)=Xu(q).X(q)=0 ( ) 2 2

0

=E q F q G qe q f q g q

( ) ( ) ( )( ) ( ) ( )

(,)=(1,0) (0,1) f(q)=0. :

33

a q a qa q a q

E qG q

e qg q

e qE q

g qG q

11 12

21 22

100

00

0

0

( ) ( )( ) ( )

( )( )

.( )

( )

( )( )

( )( )

, =

=

. : 3.4.2 S . S . . S X:US S U X(U) . q=(u,)U p=X(q)

Nu(q)=(dN)p(Xu(q))=k(q)Xu(q) N(q)=(dN)p(X(q))=k(q)X(q) k:UR . u

kXu=Nu+kXu=Nu+kXu=kuX U Xu,X ku=k U U k U. : ) k=0 U. Nu=N=0 U N=a . (a.X)u=N.Xu=0 (a.X)=N.X=0 U a.X=c X(U) a.x=c. ) k0 U. (N+kX)u=(N+kX)=0 U N+kX=b , X

k k k = =1 1 1b N X(U) x b =1 1

k k.

:(,)S , , (0)=p. N(s)=No(s) N(s).(s)=0 N(s).(s)+N(s).(s)=0. (0)=k(0)n(0) ( k ) N(0)=(dN)p((0)) (dN)p((0)).(0)+k(0)n(0).N(p)=0 :

k(0)n(0).N(p)=IIp((0)). kn=k(0)n(0).N(p)=k(0) ( (0,) n(0) N(p)) S p. . kn (0) (0). : 3.5.3 (Meusnier) S p p p. kn(w) S p wTpS.

kn(w) p S p w N(p) ( S). kn(w) S w. kn(w)>0 p N(p) kn(w)

34

knp

p( )

( )( )

www

= . . .. IIp=Ip pS2 1 ( IIp=0 p) . . wTpS TpS

w=w1+w2 kn(w)=IIp(w)=p=k12+k22 k1,k2 . :

kn(w1+w2)=k12+k22. Euler . K=k1k2>0 ( k1,k2 ) S p k1,k2 ( .. ). p . K=k1k20 ( ) p (.. ). k1=k2=0 ( ) p (.. ). . 1) . X(u,)=(()u,()u,()), u(0,2), (a,b) Xu(u,)=(()u,()u,0), X(u,)=(()u,()u,()), (u,)= 1

2 2 + ( ) ( )(()u,()u,()) E(u,)=()2, F(u,)=0,

G(u,)=()2+()2. Xuu(u,)=(()u,()u,0), Xu(u,)=(()u,()u,0) X(u,)=(()u,()u,()). :

e u f u g u( , ) ( ) ( )

( ) ( ), ( , ) , ( , ) ( ) ( ) ( ) ( )

( ) ( ).

=

+ = =

+ 2 2 2 20

3.5.1 Xu X :

k eE

k gG

= =

+ = =

+ (( ) ( ) ) , (( ) ( ) ) /2 2 1/2 2 2 3 2

Gauss :

K k k= = +

( )(( ) ( ) )2 2 2

.

35

C G=()2+()2=1 =, = + = ( ) (( ) ( ) )2 2 :

E F G e f g

k k

K H

= = = = = = = =

= =

2 0 1 0

2

, , , , , ,

, ,

, ( ) .

C Gauss 1. C

, >0, ()2+()2=1 = = K 1. ( ) ()=A A>0 (0,)

36

f(0,0)=hxy(0,0)=0 ( Xx(0,0)=(1,0,0), Xy(0,0)=(0,1,0) ) k1(p)=hxx(0,0), k2(p)=hyy(0,0). Taylor :

h x y k x k y R x y x y R x yp( , ) ( ) ( , ) ( , ) ( , )= + + = +12121

22

2

(x,y)U (0,0) R(x,y) : lim ( , )( , ) ( , )x y

R x yx y + =0 0 2 2 0 .

) IIp(x,y) (x,y,h(x,y)) TpS. ) K=k1k20 S z= TpS (. ) p xy- :

k x k y12

22 2+ =

: (i) p : k1,k2, p =0 k1,k2 S p TpS S p p ( ). .

(ii) p : 0, y kk

x= 12

=0

S p TpS ( ). (0,0) , =0 ( S TpS). . 3) . S=F1({a})={(x,y,z)W : F(x,y,z)=a} 3.1.2 ( a F). pS F(p)(0) S p :

T S F p a a a a Fp i i= = = ={ : ( ) } {( , , ): }w w 0 01 2 3 F1=Fx(p), F2=Fy(p), F3=Fz(p), F11=Fxx(p), F12=Fxy(p) ....

Gauss N= FF

w=(a1,a2,a3)TpS :

p p p

p i j i j

dF p

D FF

F p

F pD F

F pa a F

( ) ( ) ( )( )

( ) ( ) ( ) ( )

( )( ) ( )

( ).

w N w w w w w w

w w

= = =

= =

1 1

1 1

- a a Fi j i j a Fi i = 0 ( (a1,a2,a3)TpS) ai2 1= ( (a1,a2,a3) ). Lagrange ( Fi j=Fj i) a1,a2,a3,,R :

a F a F j

a F ai ij j j

i i i

= + == =

, , ,, .

1 2 3

0 12

:

37

F F F FF F F FF F F FF F F

aaa

ai

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

1

2

3

2

0

0000

1

=

=

. , .

:

P

F F F FF F F FF F F FF F F

( ) .

=

=11 12 13 1

21 22 23 2

31 32 33 3

1 2 3 0

0

R a1,a2,a3,R a1,a2,a3,, . :

a a F a a Fi j i j j j j= + = 2

F p( )

.

:

kF F F

ii i= + +=

12

22

32

1 2, ,

1,2 P()=0. K,H.

F F FF F FF F F

11 12 13

21 22 23

31 32 33

1

2

3

0 00 00 0

=

P()=0 F F F1

22 3 2

23 1 3

21 2 0( )( ) ( )( ) ( )( ) + + =

K F F FF F F

H F F FF F F

= + ++ += + + + + ++ +

12

2 3 22

3 1 32

1 2

12

22

32 2

12

2 3 22

3 1 32

1 2

12

22

32 3 22

( )

,

( ) ( ) ( )( )

./

w=(a1,a2,a3)TpS : a a Fi j ij = 0 a Fi i = 0.

w=(a1,a2,a3)R3 w.N(p)=0 ( wTpS) ((dN)p(w)w).N(p)=0 ( (dN)p(w)TpS w w ) :

a a aa F a F a FF F F

i i i i i i

1 2 3

1 2 3

1 2 3

0 = a Fi i = 0. p (a1,a2,a3)R3 a Fi i = 0.

38

.. S={(x,y,z)R3: xa

yb

zc

2

2

2

2

2

2 1+ + = } F(x,y,z)=

= 12

2

2

2

2

2

2( )xa

yb

zc

+ + =F xa

yb

zc

( , , )2 2 2 ( )/

//

Fa

bc

i j =

1 0 00 1 00 0 1

2

2

2

.

Gauss :

Ka b c

xa

yb

zc

= + + 12 2 22

4

2

4

2

42.( ) .

(x,y,z) :

a a a

a a a b a cx a y b z c

a xa

a yb

a zc

1 2 3

12

22

32

2 2 2

12

22

32 0/ / /

/ / /= + + = .

1. S X(u,)=(u2,2u u2 2

2+ 2u+,

u2+), (u,)U U q=(0,1) p=X(q)S. , Gauss S p.

2. S X(u,)=(u+u2,+u2+2,2u+22), (u,)U U q=(0,0) p=X(q)S. S p, p TpS .

3. (i) {(x,y,z)R3: xa

yb

zc

2

2

2

2

2

2 1+ + = } 0

39

N1(p) N2(p) S1,S2 : k 2 2 1

222

1 22 = + . 12. S N:SS2 Gauss p

H(p)=0. (i) p . (ii) w1,w2TpS (dN)p(w1).(dN)p(w2)=K(p)w1.w2.

13. (i) S p0S |p0||p| pS. Gauss |p0| 2K(p0)1. (ii) .

14. S N:SS2 Gauss . *(i) 0>0 0

40

3.6 . X:US S. ( Frenet) p=X(q) X(U) R3

)()()()(

)(qXqXqXqX

qu

u

=N . ,

N(q) Xu(q) X(q). '''' q U (Xu)u,(Xu),(X)u,(X),Nu,N Xu,X ,N U. N.Nu=N.N=0, N.Xuu=e, N.Xu=f, N.X=g. :

NeXXX uuu ++= 211111 NfXXX uu ++= 212112 NfXXX uu ++= 221121 NgXXX u ++= 222122

XaXa uu 2111 +=N XaXa u 2212 +=N

. e,f,g:UR S X.

kji :UR Christoffel S X. Xu X

FEXX uuu2

11111 += GFXX uu 211111 += .

uuuuuuu EXXXX 21)(

21 ==

EFXXFXXXXXX uuuuuuuuuu 21)(

21)( === '

111 112, . ( Xu=Xu) :

.)(2

2,

)(22

,

,)(2

,)(2

)(22

,)(2

2

22222

122

212

221

112

121

22

122112

22

112111

FEGFGFFEG

FEGFGGGGF

FEGFEEG

FEGFGGE

FEGFEEEEF

FEGFEFFGE

uu

uu

uuuu

+=

===

=

=

=+=

Christoffel. : '' Christoffel S ''.

41

e,f,g:UR S. , , . :

,

,

22

22

uu

uuu

XFEGgEfFX

FEGfGgF

XFEGfEeFX

FEGeGfF

+

=+

=

N

N

Weingarten. . . - X(u,)=(()u,()u,()), u(0,2), (a,b) E(u,)=()2, F(u,)=0, G(u,)=()2+()2. Eu=Gu=Fu=F=0, E=2, G=2(+) :

.)()(

)()()()(),(,0),(

,0),(,)()(),(

,)()(

)()(),(,0),(

22222

122

212

112

222

11111

++==

==+

==

uu

uu

uu

'''' .

(Xuu)=(Xu)u, (X)u=(Xu) Nu=N u . :

).()()()()()(

)()()()(

21112

12112

212

212

211

111

112

1122212

222

122

211

211

212

112

111

111

XaXafffXXXeXXXXaXae

egXXXfXXX

uuu

uuuuu

uuu

+++++++++++=++

+++++++++

NNN

NNN

X Xu Weingarten :

.)()(

,)()(

122

211

112

212

111

1122

2

212

212

211

112

222

211

212

111

212

2112

2

+==

++==

u

u

FEGfegFFK

FEGfegEEK

:

.)()(

,)()(

112

112

122

212

111

122

112

222

112

1222

2

122

211

112

212

222

2122

2

++==

+==

u

u

FEGfegGGK

FEGfegFFK

:

+

=GFG

FEE

GE

GFG

FEGF

EFEGFE

FEGK

u

u

u

uuuuu

2121

21

210

21

21

21

21

21

21

)(1

22

.

42

Gauss Gauss, , ( ) S. : Egregium (Gauss) Gauss K . ( F=0) Gauss :

+

=

u

u

EGG

EGE

EGK

2

1.

N (Xuu)=(Xu)u,(X)u=(Xu) :

.)(,)(2

12112

222

122

211

111

212

112

+=+=gfegf

gfefe

u

u

Mainardi-Codazzi ( Nu=Nu ). . (Bonnet) VR2 E,F,G,e,f,g:VR E,G,EGF2>0 V Gauss Mainardi-Codazzi V. Y:VR3 E,F,G,e,f,g. qV UV q X:UR3 E,F,G,e,f,g U. X:UR3 E,F,G,e,f,g U :R3R3 X*=X U. ( ) Gauss Mainardi-Codazzi. F=f=0 U Mainardi-Codazzi :

),(21

Gg

EeEe += )(2

1Gg

EeGg uu += .

1. Christoffel S

z=h(x,y). 2. X:US ( F=0)

Gauss : KEG

EEG

GEG

u

u

= +

12

.

3. X:US F=f=0 U. Mainardi-Codazzi e E e

EgG

= +12

( ), g G eE

gGu u

= +12

( ) .

4. X:US ,:UR = EG F 2 (u0,0)='' (0

43

u=u0 =0 S ''. (i) (log ) , u = +111 122 (log ) = +121 222 (ii) u E G E G= =

112 121 122 221, . 5. S X:US (i) E=G=1, F=0,

e=2, g=3, f=1, (ii) E=1+u2, F=0, G=1+u2+2, e=f=0, g=u2+2, U; .

6. X:R2S E=1, F=e=f=0 R2. h:R[0,+) g(u,)2=h()G(u,) (u,)R2.

7. X:US qU Christoffel Gauss S X(U) .

8. UR2 X:U->S E=G=1 F= U :U(0,) . (i) Christoffel Gauss : K u=

. (ii) e=g=0 U

C0 u=C U. 9. X:US E=1+2, F=0, G=1 e=0 U.

(i) Christoffel Gauss S. (ii) f g . *(iii) u=. S .

10. S2 . 3.7 S Gauss N:SS2. :ISR3 S T(s)=(s),n(s),b(s) Frenet . ( s) (s)=T(s)=k(s)n(s) : ) S ( (s)): ((s).N((s))N((s))=(k(s)n(s).N((s))N((s))=kn(s)N((s)) s, , ) S ( (s)): (s)kn(s)N((s))T(s)S, S, , , . (s)kn(s)N((s)) s. ( s) V(s)=N((s)) U(s)=V(s)T(s) s () T(s),U(s) T(s)S, () T(s),U(s),V(s) R3. Darboux S, S. (s)kn(s)N((s)) V(s)=N((s)) T(s) kg(s) (s)kn(s)N((s))=kg(s)U(s). kg(s) s. :

T(s)=k(s)n(s)=kg(s)U(s)+kn(s)V(s) Darboux S. ( Frenet) U.U=0, U.T=U.T=kg, V.V=0, V.T=V.T=kn, V.U=U.V g:IR Darboux S :

44

dds

k k

dds

k

dds

k

g n

g g

n g

T U V

U T V

V T U

= +

= +

=

,

,

.

g . .

: ) kn(s) s ( )

T(s)=(s), (s). kn(s)=I(s)((s)). ) g(s) s

T(s)=(s), (s). 0I p=(0)S. V(0)= d

dss d d

sp p

== =

0

0 0N N N T( ( )) ( ) ( ( )) ( ) ( ( )) g(0)=V(0).U(0)=(dN)p(T(0)).(N(p)T(0)) T(0). g(w) wTpS. w1,w2 TpS , k1 k2, w1w2=N(p) ( w1,w2,N(p) R3). wTpS :

w=()w1+()w2 w1 w ( TpS N(p)). ( N(p)w1=w2 N(p)w2=w1): g(w)=(dN)p(w).(N(p)w)=(k1()w1+k2()w2).(()w2()w1)=(k2k1).

Euler : 3.7.1 w1,w2TpS w1,w2,N(p) R3 k1,k2 . wTpS :

kn(w)=k12+k22 g(w)=(k2k1), w1 w.

=(s) n(s) V(s)=N((s)) ( T(s)) ( (s) k(s)0) :

n=()V+()U, b=()V()U V=()n+()b, U=()n()b

Frenet : kg=T.U=kn.(()n()b)=k, kn=T.V=kn.(()n+()b)=k,

g=V.U=(()n+()b).(()n()b)= =(()n+()b).(()n+()(kT+b)()b+()n)=

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

kg(s)=kn(s)=0 . : 3.7.2 =(s) n(s) V(s)

:I->S : kg(s)=k(s)(s), kn(s)=k(s)(s)

g(s)=(s)+(s) (s), . k(s)0. :

k k kg n= +2 2 .

45

S: 3.7.1 ) :IS S kg(s)=0 s. ) :IS S kn(s)=0 s. ) :IS S s T(s)=(s)T(s)S (dN)(s):T(s)ST(s)S. :

) :I->S n, , S. g(s)=(s) (s), . k(s)0. g(w) wTpS (0)=p (0)=w ( ). kn(s)=k(s).

) :IS n, , S . kg(s)=k(s) g(s)=(s) (s).

) :IS () :IR (dN)(s)((s))+(s)(s)=0 sI, :

ddsV T=

I ( Rodrigues). Darboux g(s)=0 sI. . , , . . X:US XuX N :IS (s)=X(u(s),(s)). :

E u Fu G( ) ( ) + + =2 22 1 I ( E,F,G X (u(s),(s))U). : T

T

= = + = = + + + + =

= + + + + + + + ++ + +

X u X

X u X u X X u X

u u u X u u X

e u fu g

u

uu u u

u

,

( ) ( )

[ ( ) ( ) ] [ ( ) ( ) ]

[ ( ) ( ) ]

2 2

111 2

121

221 2

112 2

122

222 2

2 2

2

2 2

2

V

kn= e u fu g( ) ( ) + + 2 22 ( ).

k u u u X u u Xg uU= + + + + + + + [ ( ) ( ) ] [ ( ) ( ) ] 111 2 121 221 2 112 2 122 222 22 2 . :

+ + + = + + + =

u s u s s u s u s s u s s u s s s

s u s s u s u s s u s s u s s s

( ) ( ( ), ( ))( ( )) ( ( ), ( )) ( ) ( ) ( ( ), ( ))( ( )) ,

( ) ( ( ), ( ))( ( )) ( ( ), ( )) ( ) ( ) ( ( ), ( ))( ( )) .

111 2

121

221 2

112 2

122

222 2

2 0

2 0

: 3.7.1 pS wTpS >0 :(,)S (0)=p (0)=w. S, c:IS c(0)=p c(0)=w c= I (,).

46

. S R3 (.. , ) pS wTpS :RS (0)=p (0)=w ( R). kg kg=T.U=T.(NT)=(TT).N : k u u u u u u X Xg u= + + + + + + [( ( ) ( ) ) ( ( ) ( ) ) ]( ). 112 2 122 222 2 111 2 121 221 22 2 N (XuX).N=|XuX|= EG F 2 : k u u u u u EG Fg = + + + [ ( ) ( )( ) ( ) ( ) ( ) ] 112 3 122 111 2 222 121 2 221 3 22 2( , , = + + 3 2 2 3 22( ( ) ( ) ) /E u Fu G ).

, . .. S xy- E G F i j

k= = = =1 0 0, , k u ug = . VT=knTTgUT=gV g=(V).V=((dN)(T)T).N=((Nuu+N)(Xuu+X)).N - Weingarten :

gEG F

u uE F Ge f g

=

1

2

2 2( ) ( )

( = + + 2 2 22E u Fu G( ) ( ) ). :IS (t)=X(u(t),(t)) ( )

( ( )) ( ) ( ) ( ( ))( ( ), ( )) ( ( ), ( )) ( ( ), ( ))( ( ), ( )) ( ( ), ( )) ( ( ), ( ))

=

t u t t u tE u t t F u t t G u t te u t t f u t t g u t t

2 2

0

tI. : 3.7.2 S pS . S p . . X:US S p k1>k2 X(U) q=X1(p). (u,)U

2 2

0

=E u F u G ue u f u g u

( , ) ( , ) ( , )( , ) ( , ) ( , )

(,)

( ) : ( ( , ) ( , ) )( ( , ) ( , ) )a u b u a u b u1 1 2 2 0 + + =

a1,a2,b1,b2:UR a ba b1 1

2 20 U.

a u du b u d a u du b u d1 1 2 20 0( , ) ( , ) , ( , ) ( , ) + = + =

47

q, 1(u,)=. 2(u,)=. 1,2:UR . a b

uii

ii

=0 (i=1,2) U

a q b qa q b q

1 1

2 20

( ) ( )( ) ( )

=(1,2):UR2 q. Y:V->S Y(s,t)=X(1(s,t)) V (q) S p. Y, s=. t=. X 1(u,)=. 2(u,)=. S. F=f=0 . :IS (t)=X(u(t),(t)) ( - )

e u t t u t f u t t u t t g u t t t( ( ), ( ))( ( )) ( ( ), ( )) ( ) ( ) ( ( ), ( ))( ( )) + + =2 22 0 tI. : 3.7.3 S pS K(p)

48

hs

s hs

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

= = =

= =

L hr s

s hs

s ds dds

hr

s s hr

s s ds

hr

s s hr

s s ds hr

s s ds

h ( ) ( ( , ). ( , )) { ( ( , ). ( )) ( , ). ( )}

[ ( , ). ( )] ( , ). ( ) ( , ). ( )

0 0 0 0 0

0 0 0

2

h(r,)=p=. h(r,)=q=.. (s)=T(s)=kg(s)U(s)+kn(s)V(s). E s

rh(r,s) S h(r,s)=(s) hr

s( , )0 T(s)S. V(s) T(s)S :

= L k s s hr s dsh g( ) ( ) ( ) ( , )0 0U

.

X:US :IS (s)=X(u(s),(s)).

U(s)=Xu(u(s),(s))c1(s)+X(u(s),(s))c2(s) c1,c2:[,]R . :[,]R ()=()=0 :

h(r,s)=X(u(s)+r(s)c1(s),(s)+r(s)c2(s)) ( >0 h |r|0 t1,t2(t0,t0+)(,) t1,t2 S (t1) (t2) . S (- ) S . - - . .

p,qS S p,q. S=R2\{(0,0)} p=(1,0), q=(1,0) ( (0,0) S). S R3 p,qS ,

49

(.. ). . X(u,)= =(()u,()u,()), u(0,2), (a,b) F=f=0, (- u=. =.) .

111 121 221 112 2 2 122 222 2 20 0 0= = = = + = =

+ + , , , ( ) ( ) , , ( ) ( )

(s)=X(u(s),(s)) ( ) ((s))2(u(s))2+(((s))2+((s))2)((s))2=1

+ =

+ + +

+ =

u s ss

u s s

s s ss s

u s s s s ss s

s

( ) ( ( ))( ( ))

( ) ( ) ,

( ) ( ( )) ( ( ))( ( )) ( ( ))

( ( )) ( ( )) ( ( )) ( ( )) ( ( ))( ( )) ( ( ))

( ( )) .

2 0

02 2

22 2

2

=0 ( (s)=X(u(s),0) u(s)) (0)=0 ( u(s)=s+ ,) 0 C z. n=No . (s)=X(u(s),(s)) s0 sns0 sns0 (sn)=0 (s0)=0 ( >0 u(s0)0) ((s0))=0 =(s0) (s0). ( 3.7.1) (s)=0 s. =0 (0)=0 (s)0 s s0I. ((s))2(u(s))2+(((s))2+((s))2)((s))2=1 + =u s s

su s s( ) ( ( ))

( ( ))( ) ( )2 0 (s)0 (

s) .

((s))2(u(s))2+(((s))2+((s))2)((s))2=1 + =u s ss

u s s( ) ( ( ))( ( ))

( ) ( )2 0 . u=. . .. ( (0,0,0)) . .

+ =u s ss

u s s( ) ( ( ))( ( ))

( ) ( )2 0 dds

s u s( ( ( )) ( )) 2 0 = ( ( )) ( )s u s c2 = . =(s) (s) Xu(u(s),(s)), (s)

= = =

( ) ( ( ), ( ))( ) ( ( ), ( ))

( ( ), ( )) ( ) ( ( )) ( )s X u s ss X u s s

X u s s u s u s u suu

u

( ( ))u s r((s)) (s) ( z) :

50

3.7.6 (Clairaut) S :

r s c( ( )) (s)= =. . c=0 . c0 u(s)0 s, u s=s(u) =(u)(=(s(u))) ( s). A ((s))2(u(s))2+(((s))2+((s))2)((s))2=1 ( ( )) ( )s u s c2 = ( ( ) ( ) )( ) ( ) ( ) + + = 2 2 2 2

4

2ddu c

u cc

d= + 1 2 22 2 ( ) ( ) ( )( ) +A A .

1. S

z=xy p=(0,0,0) .

2. S . (i) :IS . (ii) S k>0 .

3. S . :IS .

4. S S .

5. X:US ( F=0) S XuX N :I->S (s)=X(u(s),(s)). (i) =(s) Xu(u(s),(s)) (s) ( T(s)S N((s)))

kEG

Gdds E

duds

ddsg u=

+

12

. (ii) -

(kg)1 (kg)2 =. u=. . (iii) k k k

ddsg g g= + +( ) ( )1 2

( Liouville).

6. :I->S2

k s k sk s

k sgg

g( ) ( ) ,

( )

( )= + = +

221 1

. 7.

x=coshz z. 8. wTpS (0)=p (0)=w

k(0)=0. g K p( ) ( )w = . 9. :[,]R3

k>0. >0 X(u,)=(u)+b(u),

51

(a+r,0,0) (ar,0,0) . (i) . (ii) . (iii) ,

a ra r+ .

12. .

13. S1,S2 S1. S2 S1,S2 .

14. X:US S :IU Xo S ( ). Christoffel S X 221 112 111 122 222 1210 2 2= = = =, , .

15. S :S S ( ). X:(,)R3 X(s,)=(s)+V(s), V(s)=N((s)). S X Gauss .

16. * S pS . :(,)S (0)=p, kn(0)2g(0)kg(0) (0)TpS ( (0)).