CUIR at Chulalongkorn University: On the curvatures of curves in educlidean N-space

Appendix

Tl

A p p e n d ix A

The m aterial of th is part is drawn from reference Existence theorem ^Oil lin ear d iffe re n tia l equations.

A .l Theorem. The d iffe re n tia l equations

เ ^ น 1(ร) = E e * (ร)v > . i = 1 , 2 , . . . , p, p . . . - ( 1 )

where are continuous functions in the in terval 0 < s <1 r , has a set of C^-solutions which assume prescribed value น? when ร = 0.

Proof. To prove th is, le t

ui (s) " ui + j £ 1°น11(t)uk at น?(ร) = น?','

J

H c .k(t)uJ(t)dt น“ (ร) “ น " ,' ] E

'ร

\

_{/ (2)}

J

Since c^k are continuous functions on a compact set they are hounded i . e . , there exist a constant c such th at

c (ร) < c/p for a ll ธ £ [o,r] and for i = 1 , 2 , . . . , p, k = 1 , 2 , . . . , P.

IP-

Then

We assume th at 1/ \ 0 u t(ร)- น~

า าน. .i0 (c P

J

_{0 fe]}y - P _{f s} k=lE J

0 K ^{7 2} P

^ 0

■ < K for i = 1,2.

dt

พ

t,u k

cik (t) dt

dt KCs .

Also น?(ร) - น?(ร) - j E i te l and also

น?(ร) - น?(ร) É C 1

k=l 01 lcik < t)l l \ ( t ) - \ dt

dt

< (KC)(C/n ) L

J

t dt - KC2 ร2 .

In sim ilar way i t follows th at

i R / _ ’I n - l / \I ™ ท „ ท ท

|u .(ร) - น . ( ร ) I < KC £ < KC r

1 1 ท! ท!

Hence, by the W eierstrass M -test, the sequence ^ น^(ร) \

converges uniform in the in terv al 0 <: ร c r and a continuous function น^(ร) can be defined by the equation.

lim ท-*•เพ น ? ( ธ } = น ^ ( ร ) .

Also, from the uniform ity of convergence, i t follows from (²), th a t, as n ftj ,

ui (s) = u° i + 1 Y ° ik ( t) V t ) a t -

Hence “ น^(ร) = Y2 c^^(s)u^(s) and น^(อ) = u|? .p This complete the proof of the existence theorem.

À

Appendix B

The m aterial of th is part is drawn from reference £⁵].

The solution of homogeneous lin ear d iffe re n tia l equations of order ท.

The general homogeneous lin ear equation of the ท th order is L(y) = anน ) ^ + V l (x)£ ท- ï +- - '+ al (x)£ * a0<x)y ' 0 ...(1) We use L(y) to denote the resu lt of su b stitu te any any function y in the le f t member of E qu.(l). Since m ultiplying y by a constant m ultiplies each term by a constant,

L(cy1) = cL(y1), and L(cy1) = 0, i f L(y1) = 0 .

Again, replacing y by y^+ y ² replaces each term in y by the sum of two sim ilar term s, one in y^ and one in y . Hence

L(y1+ y2) = L(yx) + L(y2) and LCy^ y 2) - 0, if L(y1 ) = 0, L(y2) = 0.

Consequently the sum of two solutions, or the product of one solution by a constant, is again a solution of the homogeneous equation ( Hence we have proven the f ir s t part of the following theorem.

B .l Theorem. The solution of the homogeneous equation (¹) form a vector space พ. Furthermore dim(w) < ท.

7 5

Proof. The f ir s t part of th is theorem follows at once from the above remark. To prove the second p a rt, le t น^,น9, . . . ,น^ he k lin early independent solution of ( l) . Thus the associated vector-valued

functions บ^,บ²5. . . , บ would he lin early independent, where บ ^ x ) = (น1 (x), น/( x ) ..., น^1-1^ (x )), i = 1 ,2 ,... ,k.

To see th is , i f บ^,U g ,..., บk are lin early dependent vector-valued function, then there are scalars c ^ ,C g ,..., ck not a ll zero, so th at

c1บ1(x) + CgUgCx) + ... + ckบk(x) = 9 , where 9 = zero vector.

But th is implies

C1น1(x) + c 2น2น ) ckuk(x) = 0 , therefore น^, น

Now since บk ,U g ,..., บ are lin early independent then

บ^(Xq), บ2น 0) , . . . , บk(xç) would he lin ear independent in Vfi. But there can be no more than ท lin early independent vectors in V since dim (vn) - ท.''' Hence k cannot exceed ท and dim(w) <r ท.

Thus the theorem is proved.

2’ \ are lin early dependent.

76

A p p en d ix G

A P roof of Remark A-.1.7.

Let F ะ J ร j^o ,^lJ —^ Rn be a c ^ -p a r a m e tr iz a tio n by a rc l e n g th . Let Sq c j and assume t h a t th e v e c to r s F (ร^q),

F /;(S^q), F #(S^q) , . . . , F^2'+"^(S^q) ( r < k) a re l i n e a r l y independent then th e v e c to r s F 1 ( ร ) , . . . , F^r + ar e l i n e a r l y independent in some neighborhood TJ of Sq in J .

P ro o f. Let M ((r+ l)x ท) be the s e t o f a l l ( r + l ) X n m a tric e s w ith r e a l e n t r i e s (r+1 5Ç ท). Since th e r e e x i s t s a fu n c tio n f mapping M ( ( r + l) X n ) onto IT i n a o n e -to -o n e onto way, th e n H ( ( r + l ) \ n )

can be endowed w ith th e same topology as t h a t of r / r+ "^n . v/e denote by M ((r + l)x n, r+1) th e su b s e t of M ( ( r + l ) x n ) which c o n s i s t s of th e s e m a tric e s of ra n k r+ 1 , claim t h a t M ((r + l),y n , r+1) i s an open su b s e t o f M((r+1) x n ) . To prove t h i s we note th a t

M ((r+l)X 'n* r+1) i s a subm anifold of M ( ( r + l ) x n ) o f dim ension ( r + l ) n (see [²] on page ^{1 0 9}). Thus i t s u f f i c i e s to show t h a t i f M i s a m anifold o f dimension ท and M i s a subm anifold of M of th e/

/ / /

same dim ension then K i s open in M. Let a £ M , s in c e M i s a subm anifold of M of th e same dim ension, then t h e r e e x i s t s an open s u b s e t V 3 a of M and a fu n c tio n p such t h a t p i s a homeomorphism o f V onto some open subnet Vi of pn , and p i s a homeomorphism of

V A

M onto some open su b se t พ d V/ of R11. Because พ i s open in pn , t h e r e f o r e พ and p ^ (พ ) => V n M i s open in 17 and V

77 r e s p e c t i v e l y under th e u s u a l r e l a t i v e top o lo gy . Thus V n M te p ก V / fo r some open su b s e t p of M. This im p lies t h a t V n M i s open in M.

Hence a i s an i n t e r i o r p o in t of M , but a i s a r b i t r a r y we conclude t h a t M i s open.

To f i n i s h th e p ro o f of Remark ^ ,1 .7 » we d e fin e a new fu n c tio n

t h a t i s a ls o contimucus on J . c l e a r l y \j/'(s o) £ M ((r+ l)X n » r+1) 1 but M ( ( r + l ) x n j r+1) i s open in M ( ( r + l ) X n ) , th e n th e re i s an open su b s e t บ ³ vp (ร ) of M ((r+l),X n) and บ c M ( ( r + l ) x n , r+ 1 ).

J because "Vj/ i s con tin u o u s. Hence th e r e e x i s t s a neighborhood

u

of S^q in I such t h a t ^

(บ)

c

บ

1 so

บ

i s th e re q u ire d neighborhood#

The p ro of i s com plete.

M ((r+1)X n) as fo llow s ะ

R e f e r e n c e s

[1 J

A po stol, Tom K ., m athem atical A n a ly s is . 5 th ed.

Reading ะ Addison-W esley, 1957»

[ 2 j A uslander, L . , and Mackenzie, R.E. I n t r o d u c ti o n to D i f f e r e n t i a b l e M an ifo ld s. ;:c G raw -H ill, ^{1 9 6 3}.

^3 j B i r k o f f¹ G. and Mac L ane,ร . , A Survey o f Modern A lg eb ra. 3 rd . New York : M acmillan, ^{1 9 6 5}.

[ k ] Buck, r u e ., Advanced C a lc u lu s . 2nd, ed. New York ะ M acGraw-Hill,

1 9 6 5.

^ 5 j Gluck, Herman, H igher C urvature of Curves in E uclidean S pace.

The American M athem atical Monthly, V o l.7 3 | N o.7, ^{1 9 6 6}.

[ 6 j Kaplan, W ilfre d , and Lewis, Donald J , C alcu lu s and L in ear A lgebra New York ะ John W illey and Sons, I n c . , ^{1 9 7 0}-^{1 9 7 1}»

£ 7 ] Komogorov, A.N. and Fomin, S .V ., I n tr o d u c to r y Real A n a ly s is , t r a n s i , and ed. by S ilverm an, R ichard A. Englewood

C l i f f s ะ P r e n t i c e - H a ll , ^{1 9 7 0}.

w illm o re, T , J . , D i f f e r e n t i a l Geometry. Oxford U n iv e rs ity P r e s s , ^{1 9 5 9}.

s t r u i k , D .J ., L e ctu re s on C l a s s i c a l D i f f e r e n t i a l Geometry.

Reading : Addison-V/esley, 1950.

[

c h e r n ,■ ร.ร., Notes on D i f f e r e n t i a l Geometry. บ, o f C a l., B erkeley l e c t u r e n o tes

/

Name Degree S c h o larsh ip

79

VITA

Mr. P iro j S attayatham

B.A. (Thammasat U niversity), ¹⁹⁷¹+.

U niversity Development Commission (U .D .C .)

Thai Government, ¹⁹⁷^-¹⁹⁷⁶.