S(HAU M'S O UfLJNE

MARTI,'II ,\ LlPSC HUTZ ,.. ",

fJoII'O}-,\AllfA'Jl ~IOI,,"l O\~lTOIi,."'l ,,,,()f() lO '1I1\1Kon .. ~

f][PJ[nl ' ~I: .. '" l~; \ 8 I\I"

\j ,R\\I'-ttllf "HI'YOR~ [ 111. I \

-~

SCHAUM'S OUTLINE SERIES

-

(Schaum's Outline of Theory and Problems of DIFFERENTIAL GEOMETRY)

MARTIN . LLIPSCHUTZ, Ph. D. PROFESSOR OF TICS UNIVERSITY OF BRIDGEPORT

:

-

:

'' ,

,

McGRAW-HILL, NEW YORK ,

---~ ~-- ~ ~-

.

Copyright 1974 by McGraw-Hill, Inc. rights reserved. Printed the United States of America. part of this publication may be reproduced, stored a retrieval system or transmitted, any form or by any means, electronic, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher.

37985

'514. f

Copyright 1981, , . & , . , . " . ' , ,

(, , , , .) .

, SCHAUM 22

.

.

. 2. . 3, 4 5 3 K~K

. . o~

. 6 7 : : . 8 , 9. , ,

. .

' , .

d, Martin Silverstein Jih~Shen Chiu . . Daniel Schaum Nicola Monti Henry Hayden . Sarah .

Bridgeport, Conn. 1969 MARTIN . LIPSCHUTZ

, . . , .

. ' , . '

, , . ' , ,

. , . , ,

. , ( )

' . ' ,

. , .

.

, , ( ), Cartan Frenet

: . Do Carmo (Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976) . O'Neill (Elementary Differential Geometry, Academic Press, New York, 1966).

. .. .

. . .. , - , .

1982 .-.

~~ \i~ ~~

8 .......................................... ' 150 . . ' . , . .

9 .......................... 171 . . . . . Gauss . . Rodrigues. , -. .

10 . .................. 201 Gauss-Weingarten. Gauss. . .

. . . . .

11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 227 . . . . . . . . . -. . Gauss. -

Gauss-Bonnet.

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

................................ 264

.................................................... 267

273

1

. . 3.

, , .

. ' . .

1.1. " Q R , , CQR , Q R, . -. CQR Q R . , C r . C r . , r .

. 1-1 .I-2

1.2. Moebius (. -2) . l , " .

.. , Moebius .

. ' 3.

3 8 = (, 2, a3), a,~, .

3 8, b, C, , , ... , Q, R, . . . a -a, -a = (-a, -a2, -3).

= (, , ). ' 8 = (1, 2, 3) lal = V i + a~ + ~. 181 ~ , 181 = , 8 = .

1

2 . .

a = ( , 2, aa) b = (b t , b2, ba) Ea

a + b = ( + bt , 2 + b2, aa + ba) a b a - b = a + (-b). 1.1

[] a + b = b + a ( ) [2] (a + b) + c = a + (b + c) ( )

[3] + a = a a [4] a + (-a) = a 1.3. " a = (, -2, ) c b = (, , ).

a + b = (,-,), -a = (-,2,0), b - a = (-, 3, ), IaI =...;5. 1.4. [] c [4 ] a c b

a + (b - a) = a + (b + (-a = a + (-a) + b = 0+ b = b c a + = b , = b - a. Aoc oc. , C , a + = b,

(-a) + a + = (-a) + b = b - a + = b - a = b - a . Q Ea ( Q), PQ Q - , , PQ Q, . 1-3. Q IPQI. PQ = -QP, IPQI = IQPI, PQ = P'Q' Q - = Q' - ',

= . . I-3

1.5. " a = PQ, b = QR c c = , d = SP, . Q b . 1-4. ~H

a+b = PQ+QR = Q-P+R-Q = R-P = PK~ c a+b+c = + = R-P+S-R = S-P ----..---- S

= PS = -d

a + b + c + d = PS + SP = S - + - S = . 1-4

' k a = (, a2, aa) ,

ka ka (kat, ka2, ka3) Oa = kO = k a.

. ka .

1.4

[ ] k t (k2a) = (k t k 2)a = k t k2a (k t + k2)a = kta + k2a k(a+b) ka + kb ( ) [3] 1a = a

, a = (a, a2, aa),

.1 3

Ikal = V(kl)2 + (k2)2 + (k3)2 = , k a Ikal = Ikllal (1.1) 1.6. - a = (1,,,.,0) b = (0,2,-1). - 2a = (2,2".,0), (-1)a = (-1,-".,0) = -a a - 3b = (1, ". - 6, 3). 1.7. , u 2, U3 a = - 2U2, b = -U2 + 2U3 c = + U2 + U3. -

a - 2b - c = ( - 2U2) - 2(-U2 + 2U3) - ( + U2 + U3) = - 2U2 + 2U2 - 4U3 - - U2 - U3 = -U2 - 5U3 ' a b, k ~ a = kb. . a

b, (.) lal = Ikllbl = klbl = Ibl. k = 1 a b. , , . . a = kb, b ~ k ~ , a

b. , a = b = a ( ) b, a = kb k, a b .

' U . , Ua a. , a 1/lal,

U a = a/lal (1.2) 1.8. a = (1, -1, 3), b = (2, -2, 6) c = (-3,3, -9). a = !b, a b { . b = -(2/3)c, b " c {

. a { u. = a/lal = (1/v'll, -1/v'll, 3/v'll). 1.1). (. 1-5) a = , b = 08 .

a b : a + = a + !

a + !(b - a) = a + !b - !a !a +!b

~M O~B

. 1-5

. Ut, U2, , u .. , ( ) k t , k 2, , k .. {

kIuI + k2u2 + ... + k .. u.. = (1.3) Ul, U2, , u" ., . , Ut, U2, , U" , (1.3) k t = k 2 = ... = k .. = . ,

, , 10 + OUl + ... + OU .. = .

1.10. a = (1, -1, ), b = (0,2, -1), c = (2, , -1) , 2a+b-c = .

1.11. ' a { b. a = b = a = kb, a - kb = , a, b . . , a b { . kta + k2b = , ~ {

, ~ kt"'" . a = -(kJkt)b. . , { , { .

4 : 1

I.lO : .. ' , . , Ut, U2, ... , Un

U = ktut + k 2u2 + ... + knu .. k t = k;, k2 = k~, ... , k .. = k~.

el = (1, , ), e2 = (, 1, ) e3 = (, , 1) . , ktet + k 2e2 + k 3e3 = (kt, k 2, k 3), ktet + k 2e2 + k 3e3 = k t = k2 = k3 = . ' , a = (, a2, 3) a = alel + a2e2 + a3e3, et, e2 e3, .

. , 3, (i) 3 ()

. . : .2. 3. ' , 3 .

' Ut, U2, U3 3 a = alUl + a2U2 + a3U3. a, 2, a3, ;, i = 1,2,3, a Ut, UZ,.U3.

, . . , , . '

, , , . , a, b, , , u, ... , bi, , , 14, .

1.12. " , U2, U3 a = 2 - u2' b = U2 - 2u3, C = 3 + u3' a, b, c . ,

kta + k2b + k3c = (2kt + 3k3)ut + (-kl + k2)U2 + (-2k2 + k3)U3 = ,

2kt + 3k3 = , -kl + k 2 = , -2k2 + k 3 = . k t , k 2, k 3

(

2 det -1 1

-2

.

k t = k 2 = k 3 == . , a, b, c . a, b, c .

: .3. " Ut, U2, U3 3

al1U! + a2lU2 + a3lU3 V2 = al2U! + a22U2 + aS2U3 Vs = al3U! + a23U2 + a33U3

,

. 5

3 V; = a;;ui, j = 1,2,3. , V2, V3 ,

;=1

( det 2 3

a = (, 2,3) b = (b t , b2, b3)

, a = b,

1.14 : [Cl] a b = b a ( ) [C2] (ka) . b = k(a b) (k = ) [C3] a (b + c) = a b + a c ( ) [C4 ] .

() a a ~ a (ii) a a = a = .

(1.4)

, a = a. . , a b = a, b b = [C4](ii) b = .

1.13. " a = (-2,1, ) b = (2,1,1). a b = -3 a a = 5 = lal2 1.14. UI. u2 a = UI - U2. b = 2uI + u2'

a b = (UI - U2) (2 + U2) = 2 - 2ul U2 + U2 - U2 U2

1.16 , Cauchy-Schwarz

Ja bJ ~ JaJJbJ , a b . a b, = ~(a, b),

a' b = JaJJbJ cos ~ ~ 7.

1.15. ABC (. 1-6) a = BC, b = AC, c = = a - b (J = 4ACB = 4(a, b). '

Icl 2 = la - bl2 = (a - b) (a - b) a' a - 2a b + b b

(1.5)

. 1-6

~E b . . () a b, Pb (a), Pb (a) (a' b)/Ibl. Pb (a)ub' Ub

b, a b Pb (a).

6 .

(a' b)b \bj2 (1.6) P b () = Pb () = . a """ , (1.5) P b (a) = lal cos Pb (a) = lal cos Ub, = 4(a, b). , Pb (a) Pb (a) b, , . 1-7. Pb (a) b,

a'(-b)(_b)

l-bl2

Pb (a) , b.

b b

. 1-7

a b a.l b, a' b = . , (1.5) a b , a = , b = , = 4(a, b) = 70/2.

1.16. " a b c = a - Pb (a). c b. , c = , (1.6) ~o = 1a - Pb (a) = la - kb, k = {a' b)/lbI2 , a, b . c #- . .

= (a - (a'b)b) b = a' b _ ..:....{a_.-.,b;'7){"...b_b....:..) Ibl2 Ibl2 (a' b) - (a' b) = c.l b.

W el, ez, e3 , . 1-8. . , kIeI + k~2 + kaea = , = ej' = ej' (kIeI + k~2 + k3e3) = ej' kIe; = k i , k ; =

. , . ei, = 1,2,3, ,

el el = e2' e2 = e3' e3 1 = e2' ea = el' ea =

( ) ( ) , ,

{1, j =

ej' ej = ] = , j -F (i,j = 1,2,3)

. l-8

(1.7)

] ( , j) Kronecker Kronecker .

.1

1.23 : 1.4. eI, e2, e3 a Qlel + Q2e2 + Q3e3, b b2t'2 + b3e3 . 3

() a . b = Q 1bI + Q 2b2 + Q 3b3 = ab; =1

() lal = ya:a = yI~ + ; + ; = ~ ~ af () a = a et, ( = 1,2,3). 1.17. a = eI + 2e3, b = 2eI + e2 - 2e3 c = -2e2 + e3'

() a b = (1)(2) + (0)(1) + (2)(-2) = -2 (b) (a c)b = [(1)(0) + (0)(-2) + (2)(1)] (2eI + e2 - 2e3) (c) lal v'12 + 22 = V5 (d) u. = 1:1 = (l/V5)eI + (2IV5)e3

a .b -2 cos 4-(a, b) = - -11-1 = -

a b 3/5 (e)

a = QIeI + 02e2 + Q3e3 = 4-(a,et), = 1,2,3, . 1-9. cos , cs2, COS 3 ( ) a. a e; = ial cos = ,

cos = at/lal, = 1,2,3 V

a 2 3 \a\ =\a\ eI + r; e2 + \al e3 (cos )e + (cos 2)e2 + (cos 3)e3

. 1-9

7

bIeI +

a a.

(eI, e2, e3), (gl, g2, g3) (gl, g2, g3) , gl g2 el e2 . g3 e3, (gl, g2, g3)

(el, e2, e3), g3 e3, .

, ( ). 3

(Ul, U2, a) (, 2, 3) ; = ajUi. =1 (, V2, 3) ' (Ul, Uz, ns), det (ai) > . 1.27 3. . ,

. . , , (Ul, U2, a) , ,

, .

8 .

1.18. (u l U2' U3) . 1-10() (c) { . . 1-10(b) (d) { .

() (b) (c) (d) . - . , , .

' (et, e2, e3) a = alel + a2e2 + a3e3, b = btet +

b2e2 + b3e3. a b, aX b,

aX b = (a2b3 - a3b2)el + (3b l - at b3)e2 + (a t b2 - 2b t)e3 ,

det (: :: \e3 3

axb b l ) b2

1.19. ' a = e l - e2 b = e2 + 2e3'

(

e l 1 a b = det e2-1

e3

1.32 . , 1.31 :

1.5. () la bl = lallbl sin , = 4-(a, b) () . (a b) . a (a b) . b

b. aX b 7'= , (a, b, aX b) ().

, lallbl sin = lal = , Ibl = , = , = 71", () Schwarz ( la bl = iallbl a b ) :

1.6. a b = , a b .

.,

. 1

' a b , a b oF

1.5() aX b - aX b a b (a, b, aX b) , . 1-

II(). . , b a aX b [. 1.5

()] [. 1.5(ii)a], [. 1.5(ii)b]. ' b a = -(a b), . 1.11(b).

1.20. (, g2, g3) (. 1-12) 1.5

= g2 = g3

g3 = -gz

gz = -g3

gz gz = gz g3 =

(3 gl = g2 g3 gz = -gl g3 g3 =

9

() (b) . -11

. I -12

1.29.

, -

[] aX b = -(b a) aX (b + c) = aX b + aX c (ka) b = k(a b) aX a =

( ) ( ) (k = )

' . " , a (b c) oF (a b) c.

, 1.20 gl (gI gz) = gI g3 = -gz, (gI gI) gz = gz = . 1.21. ABC (. 1-13). " a = BC, b = AC, c = = b - a, = 4-(b, c), = 4- (c, a) = 4- (a, b). '

= c c = c (b - a) = c b - c a cXb = cXa

', cxb (b - a) b = b b - a b = bXa

cxb = cXa = bXa

, Ic bl = Ic al = Ib al Icllbl sin = Icl !al sin = Ibllal sin '

sin lal

sin Ibl

sin Icl

. 1-13

a' b c . . , a' (b c), a b c. . ,

2

10 . 1

( , bt ,)

a o b c (alel + a2e2 + a3e3) det e2 b2 C2 e3 b3 C3

al(b2C3 - b3C2) a2(c3bl - ctb3) + a3(blC2 - b2ct) (108)

' aobc = 'ab = boca = -(boac) = -(coba) = -(aocb) (1.9) a b c = aX b c. ,

. . [abc] = a b c = a b c

n 1.3 (1.8) : 1.7. [abc] = , a, b, c .

. . , 1.35, : 1.80 a (b ) = (a c)b - (a b)c

" : [Ft ] (ab)o(cd) = (aoc)(bod)-(aod)(boc) [F2] (a b) ( d) = [abd]c - [abc]d

1.22. " u = c d. a b u = a b u = a [b (c d)] = a [(b d)c - (b c)d] (1.9) 1.8.

(a b) (c d) = (a c)(b d) - (a d)(b c) [Ft ].

101. [] [] . , ,

[] a+b = b+a, [A2](a+b)+c = a+ (b+c), [A3]a+O = a, [A4]a+(-a) = . []: a + b = (

.

1.2. . 1-14 a = , b = , c = OS. , VQ a, b

c.

1.3.

" OV + RV VQ VR+ RQ

RS + 8

+ 08 = b + c -RV + RQ -08 + = -c + a + OS + 8 -b + c + a

. 1-14

[] [4] (. 1.4) a + = b , = b + (-a) = b - a.

, : () , ' + a = a, ' = . (b) -a , a' + a = , a' = -a. (c) - (-a) = a a. () + a = a.

(b) + a = .

11

u

-

(c) -a + = . . = 0- (-a) = -(-a). . = a, -a + a = -(-a) = a, .

l. 1.;: 1.4. [ ] [3],

. , , [ ] k t(k2a) = (ktk 2)a' [2] (kt + k 2)a = kta + k2a, k(a + b) = ka + kb [3] la = a.

(k t (k2a t), k t(k2a2)' k t(k2a3 = ktk 2)at, (ktkz)az, (k lk 2)a3) = (ktkz)a

k} + k 2)at , (k} + k 2)a2, (k} + k2)a3) (ktat + k2at, kla2 + k2a2, kla3 + k 2a3)

k(a + b) = (k(a} + bt ), k(Z + b2), k(a3 + b3 (ka} + kb t , kz + kb2, ka3 + kb3)

[3]: la = (la, 1a2, 1a3) = (a, , a3) = a ka + kb

1.5. ' a = Ul - 2U2 + 3U3, b = U2 - U3 c = Ul + 2U2, 2a - 3(b - c) Ut, U2, U3.

" 2a - 3(b - c) = 2a - 3b + 3c = 2( - 2U2 + 3U3) - 3(U2 - U3) + 3( + 2U2) = 2 - 4U2 + 6U3 - 3U2 + 3U3 + 3 + 6U2 = 5 - U2 + 9U3

1.6. , , .

" ' AC ABC (. 1-15). = iAB, ' = iAC ' = ' - = i(AC - ) = iBC. " ' ! BC ! BC.

. I -15

c

12

1.7. ' a = , b = , b # a c = OC (. 1-16). C L, , c = kla + k 2b k l + k2 = 1.

.

L

' C L, = a - b BC = c - b .

k

o~~--------~--------~C . 1-16

c-b = k(a-b) c = ka + (1 - k)b k l + k z = k + 1- k = 1. . , c = kla + k2b k l + kz = 1, b " a,

c - b = kla + k 2b - b = kja - (1 - k2)b = kja - kjb = kj(a - b) c - b = BC a - b = C .

1.8. Ut, U2, , U n ,

. . Uj U2' ... , Un, Uj = k2U2 +

... + knun. Uj - k2U2 - ... - knun = , 1 uj . Ul,"" u n .

', Uj"", u n , ( ) k l, ... , k n kjul + k2U2 + ... + knun = .

" kl"p Uj = -(k2/kIiU2 - ... - (kn/kl)un, U2, ... , ,

1.9. , , .

' , U2, ... , Uk , U2'" .,Uk, Uk+l,"" Un . () kl, ... , kk, , klul + k2U2 + ... + kkUk = . .

klul+k2U2+ ... +kkUk+ OUk+l+ ... +Oun = , U2, , .

1.10. 1.1: ' Ul, , U n

klul + k 2u2 + ... + knUn = k~ul + k~U2 + ... + k~un k l = k~, k 2 = k~, . , k n = k~.

' j kj"p k;. (kl - k~)ul + (k2 - k;)u2 + ... + (kj - k;)Uj + ... + (kn - k~)un = k j - k; . ' Ul"'" un , .

1.11. 3

' 3.

' a, b c xa + yb + C = , = = = . ',

,

. 13

+ yb l + zCl 2 + yb2 + ZC2 3 + yb3 + ZC3 , = = = ,

,

det (:~ :~ :~) 3 b3 C3

"'"

' , u = (Ul, U2, U3) ) 3, + yb l -1- ZCl Ul 2 + yb2 + ZC2 U2

3 + yb3 + ZC3 U3 , = k l , = k 2, = k 3 , u = kla + k2b + k 3c.

1.12. 3 . , U2, U3' U4' ... , U n , .. , U2' U3, . , ' , U2' U3' U4' ... , U n (. 1.9). ' , u2' U3 , U4 = ktut + k2U2 +k3U3,

, U2' U3' U4 . . , u2' u3' U4' ... , Un .

1.13. ' Ul, U2, U3 a = Ul - U2 + 2U3, b = U2 - U3, C = -U2. 2a - b - 2c Ut, U2, U3.

2a - 11 - 2c 2( - U2 + 2U3) - (U2 - U3) - 2(-U2) 2 - 2U2 + 4U3 - U2 + U3 + 2U2 = 2 - U2 + 5U3

. 2a - b - 2c , U2, U3 2, -1, 5.

1.14. [Cl] [C4 ] 5,

. [C t]: a' b = atbl + a2b2 + a3b3 = btat + b2az + b3a3 = b a [C2]: (ka)' b = katbt + ka2b2 + ka3b3 = k(atbt + a2b2 + a3b3) = k(a' b) [C3]: a' (b ~) = at(b t + ct) + az(b2 + C2) + a3(b3 + C3)

= atbt + zb2 + a3b3 + alCl + a2C2 + aaC3 = a,b + a'c [C4]: a'a = ; + ~ + ; "" , a'a = ~ + ; + ; = a = 2 = a3 = .

1.15. (. 1-17) a = b = . ' \\ = 2, \\ = 3 = 300,

() a' b, (b) . (b), (c) . (b). () a' b JaJJbJ cos 4--(a, b)

(b) . (b)

(c) . (b)

= (2)(3) cos 300 = 3V3

(a'b)/JaJ = 3V3/2

. (b) 1:1 = (3V3/4)a . I -17

14 .

1.16. Cauchy-Schwarz la bl ""'" lallbl. . a b .

. , a b . a, b -# . [C4 ]

~ (~a ~b)' (~a ~b) 21allbl 2a'b 2a' b ~ 21allbl la bl ~ lallbl.

., ' - b . r;b ". b r;b '" .. ... . V"j;a - b = aa + bI = , a b .

1.17. la bl ""'" lal + Ibl. ' la bl2 = (a b) (a b) = lal2 + Ibl2 2(a b) ~ lal2 + Ibl2 + 21allbl

.

1.18. Ilal - Ibll ""'" la bl a b. ,

lal = la b ::;: bl ~ la bl + Ibl

' Ibl = fbf = la b - al ~ la bl + la[ lal - Ibl ~ la bl

Ibi - [al ~ [a b[ ila[ - Ibfl = Max (jal-Ibl, [bl-Ial) ~ la b[, .

1.19. ' c a b. c kta + k 2b k t , k2

, c a b C' a = C' b = . C' (kta + k2b) = kt(c' a) + k 2(c b) = " c kta + k2b.

1.20. " Ut, U2, U3 . a = Ut, b = U2 - . (U2), C ~ U3 - . (U3) - P b (U3) a, b, c .

" a' b a' (U2 - . (U2)) = a' [U2 ~ (a' u2)a/laI2]

a 1- b. ' a' U2 - (a' U2)(a' a)/laI2 = a' u2 - a' u2 =

a' c = a' [U3 - P.(U3) - P b (U3)] = a' [U3 - (a' u3)al.lal2 - (b' u3)b/lbI2] = a' u3 - a' U3 - (b' U3)(a' b)/lbI2 ' a' b = , a' c = a' u3 - a' u3 = a 1- c. '

b' c = b [U3 - (a' u3)a/laI2 - (b' u3)b/lbI2] = (b U3) - (a u3)(a b)/laI2 - (b' U3)(b' b)/lbI2 = (b' U3) - (b' U3) =

a, b, c . ' . a = u i -# . , b = ,

= b = u2 - . (U2) = U2 - ka = u2 - kut ), UI U2 . , c = , = c = U3 - . (U3) - Pb (U3) = U3 - kta - k 2b = U3 - klul - k 2(U2 - kut) = U3 - k 3u l - k 2U2 , , U2, u3 .

..,

.1 15

1.22.

{1 j = i j = ' . .' 2 21 + 322 + 423 = 2(0} + 3(1) + 4(0) = 3. 1 -#1

3

(b) , . ' ~ b = .. b = b ;= 1 ) 1 , t

3 3 3

Ul, U2, U3 V; = ;;Ui, U; = bijVi. aikbkt = 81;. 3 3 = 1 = 1

k=I

U; = ~ ; = ~ bkjVk, i k. =1 k=

, 3

Vk ~ aikui' ' = 1

3 3 3

~ ; =l

~ bk; ~ aikUi k=1 =1

3 , . u2, U3 { , j = : aikbkj'

k=1

1.23. ' 1.4: ' et, e2, e3 a = atel + azez + a3e3, b = bI el + b2e2 + b3e3,

() ab=lbl+zb2+3b3, (b) lal=Va~+;+a;, (c) i=a'ei, i=I,2,3. () a' b (alel + aZe2 + a3e3) (b1el + b2e2 + b3e3)

atbl(et et) + a 1b2(el e z) + a l b3(e l e3} + azb l (e2 el) + a2 b2(e2 e2} + a2b3(e2 e3}

+ a3bl(e3' el} + a3b2(e3' e2} + a3b3(e3' e3} alb t + a Zb2 + a3b3

a' b = (.: aiei) (.:i bje;) ,= ,=

3 3 a b = :: abjij

= 1 ;=

(b) IaI = va:a va2 + a 2 + a2 1 2 3 (c) a' ei = (1 a,-e; ) e; = :r a;(e; e;)

3 3 ~ ~ ajb/ej' ej} = =1

1.24. ' a = el - 2e2 + 3e3 b = e2 - e3. : () a' b, (b) lal, (c) u., (d) . (b), (e) . (b), () cos 4-(a, b), (g) a' el, a' e2, a' e3, (h) a. () a' b = (1)(0) + (-2)(1) + (3)(-1) = -5 (b) IaI ...;;::-; = (1)2 + (-2)2 + (3)2 = 4 (c) U. = a/IaI = (1Iy'i4)(et - 2e2 + 3e3) (d) . (b) (a' b)/la! = -54 (e) . (b) = . (b)u. = -(5I14)(et - 2e2 + 3e3)

() cos 4-(a, b) = (a b)IIa! Ib! = (-54..[2) = -5I(2V7) (g) a' r = 1, a' e2 = -2, a' e3 = 3 (h) co' a, el) = alIIa! = lIV14, cos 4-(a, e2)

16 . 1

1.25. ' ul, U2, U3 . Ul = 1

2 Ul 3 Ul

, 2, 3,

U2 2 U2 1 3 U2 VlU3 2 U3 = 3 U3 1 Uj = 8, , j = 1,2,3. ' , 2, 3 Ul, Uz, U3. ' , a = alUl + a2U2 + a3U3 b = blVl + b2V2 + b3V3,

a b = (~aiUi) (~b;Vi) = ~ aIbj(UI ) = ~ aibj 8jj = aib; , t , t )

, . ' el' e2, e3

allel + al2e2 + al3e3 U2 a2lel + a22e 2 + a23e 3 U3 a3lel + a32e 2 + a33e 3

Xlel + XZe2 + X3e3

+ 22 + 33 1 u2 = 2 + 22 + 233 ' u3 3 + 322 + 333 ' 2. 3' det () # ,

= xlel + X2e2 + X3e3' 2 3' ' , v2, v3 . -

3 klvl + k2V2 + k3V3 : kjv;

=

, " Uj, j = 1,2,3,

[ :i k.V,] u ::::::: 1 J 3 : kj(Vi' Uj) = 3 : ktJ =l 11 , j = 1,2,3 kt = k 2 = k 3 = , v2' V3 . .

1.26. (, 2, 3), = 2ul - U2 + 2U3, 2 = U2 + U3 3

2U2 + U3, (ul, U2, U3).

' ! det (-~ ~ - ~ ) = 1 > . 2 1 1

-Ul+

1.27. FJ3. : () ' (, 2, 3) (, 2, 3) (, 2, 3). (b) ' (, 2, 3) (Ul' U2, U3),

(Ul, U2, U3) ' (, 2, 3), (c) ' (WI, W2, W3) ' (, 2, 3) (, 2, 3)

(Ul, U2, U3), (Wl, W2, W3) ' (Ul, U2, U3).

,.

. 17 3 () ; = ~ j, j = 1,2,3. det () = 1, (, 2, 3)

= (, 2' 3)'

(b) - ; = ~ aijU; Uj = ~ bijVj, ' 1.22, det (: aikbkj') = det () det (b ij),

k=I

det (j) 1 det (bij) = det (aij) det ()

, (, 2' v3) ' (ut, U2' U3), det () > . det (b ij ) > (ut, u 2, U3) (, 2, 3)'

3 3

(C) ' Vk = ~ aikU; Wj = ~ bkjVk' = k=I

3

: CijUi, i=l

3 Cij = ~ aikbkj, i, j = 1,2,3. '

k=I

(wt, W2' w3) ' (, 2' 3) (, V2' 3) ' (Ul, U2, U3), det (b jj ) > det (aij) > det (Ci;) > . (Wl> W2, W3) (Ut, U2' U3)'

1.28. - a = 2et - e2 + e3, b = el + 2e2 - e3, C = e2 + 2e3.

() aX b, (b) b a, (c) aX (b c), (d) (a b) c, (e) (a b) c, (/) aX (b + c) - aX b - aX c.

() aX b det (:~ -: ~) e3 1-1

-el + 3e2 + 5e3

(b) b a (e l 1 2)

det e2 2-1 e3 -1 1

(C) aX (b c)

(d) (a b) c =

(-1 2) (2 1) ( 2 = el det - e2 det + e3 det 1 1 -1 1 -1 - ~)

aX b = -(b a).

(

e l 2 5) = det e2 -1 -2

e3 1 1

= det (:: e3

-1 3 5 aX (b c) -# (a b) c.

(-1)(0) + (3)(1) + (5)(2) = 13

18

() aX (b + c) = det(:: -~ e3 1

4e2 + 2e3' a (b + c) - a b - a c

.

1.29. () aX (b + c) = aX b + aX c, (b) (ka) b = k(a b). a = uIe} + ~e2 + ae3, b = btel + b2e2 + b3e3, c = cle} + C2e2 + C3e3' () aX (b + c) = [u2(b3 + c3) - u3(b2 + c2)Jel + [u3(b } + ct ) - u l(b 3 + c3)]e2

(b) (ka) b

+ [ul(b2 + c> - u2(b! -- Cl)Je3 (u2b3 - u3b2)et + (u3 b } - ulb3)e2 + (UI b2 - ~bI)e3

+ (U2C3 - u3c2)el + (u3c t - utc3)e2 + (Ul C2 - u2cl)e3 aXb+aXc

(kU2b3 - kU3b2)et + (kU3bl - ku tb3)e2 + (ku Ib2 - kU2bt)e3 k[(uzb3 - u3b2)et + (u3bl - u t b3)ez + (ulbz - u2bl)e3] = k(a b)

1.30. la bl2 = la[Z [b[2 - [a b1 2 a = ulel + uZe2 + U3e3 b = ble} + b2e2 + b3e3.

la bl2 = (a b) (a b) = [(U2b3 - u3b2)et + (u3b } - ulb3)e2 + (lb2 - u2bl)e3] [(u2b3 - u3b2)et + (u3bl - u lb3)e2 + (u Ib2 - u2bt)e3]

(U2b3 - U3b2)2 + (U3bl - UIb3)2 + (Ulb2 - U2bl)2 2 2 2 2 2b2 2 bZ 2 2bZ u2b3 + u3b2 + U3 + 3 + b2 + U2

- 2U2b2U3b3 - 2ulbIU3b3 - 2utbtuzb2 (a a)(b b) - (a b)(a' b)

2 2 2 2 2 . ( + U2 + u 3)(b} + b2 + b3 ) - (tb} -- u2b2 + 3b3)2 2b2 2b2 2b2 2 2 2 2 2 2 b b b b 2 + 3 + U2 + u2b3 + u3bl + u3b2 - 2ulbI2b2 - 2 3 3 - 2U2 2U3 3 .

1.31. ' 1.5: () la bl = lallbl sin , = 4-(a, b) () . (ab).a (ab).b

b. ' (a b) =F , (a, b, aX b) (et, e2, e3). ()

la bl 2 = lal2 Ibl2 - la bl2 = lal2 Ibl2 - lal2 Ibl2 cos2 lal2 1bl2 (1 - cos2 ) = lal21bl2 sin2 (Iallbl sin )2

. sin "" ~ ~ , la bl = lallbl sin . () . a = ulel + ~e2 + u3e3 b = blel + b2e2 + b3e3'

(a b) a = [(~b3 - u3b2)el + (u3bl - ulb3)e2 + (ulb2 - u2bl)e3] (ule} + a2e2 + U3e3)

b.

= UI~b3 - Ula3b2 + u2U3b! - U2utb3 + U3ulb2 - U3U2b! = ~Oo, (a b) b = . (a b) .1 a (a b) .l.. b . . (a, b, aX b)

( b} (U2b3 - U 3b2) det U2 b2 (U3b} - atb3) = (U2b3 - U3b2)2 + (U3b} - utb3)2 + (Ulb2 - u2bt)2 U3 b3 (ut b2 - ~bl)

= la bl2

' aX b # , la bl2 > (a, b, aX b) (eI, e2, e3)'

.,

. 19

1.32. .

" a b . a b .

, 1.6 c = = , . 1.5() (a, b, )

= aa + b + . ' 1.5() a = alal2 + (a b) = b = a(b a) + bl 2 = . ' a b { , lal2 1bl2 - la bl2 .,. . .

= = = . (a, b, ) (a, b, ) , > . ' 1.5() l,l = ll =

l,l. = 1 c = '.

1.33. " a = el + 2e2 - e3, b = -el + e2, c = -e2 + 2e3. a b c.

a b ( 1 -1 )

det 2 1-1 -1 2

5

1.34. a b c = a b c.

aX b

1.35. 1.8: aX (b c) = (a c)b - (a b)c.

aX (b c) (atet + a2e2 + a3e3) [(b2C3 - b3c2)et + (b3cl - c3bt)e2 + (blC2 - b2ct)e3] (a2 bl c2 - Zb2ct - 3b3C! + 3b tc3)et + (a.3b2C3 - a3b3c2 - albtc2 + alb2c t)e2 + (lb3Cl - atb t c3 - a2b2c3 + a2b3( 2)e3 (a c)b - (a b)c (alcl + 2C2 + a3(3)(btet + b2e2 + b3e3) - (tbl + a2b2 + a3b3)(ctel + C2e2 + C3e 3)

(a2btc2 + a3btc3 - a2ctb2 - a3ctb3)et + (b2a l c t + b2a 3c3 - C2albl - C2a 3b3)e2 + (b 3a lcl + b3ZC2 - C3a l b l - c3a2bz}e 3 aX (b )

' 1.36. OPQR . 1-18 a = , b = OQ, =

RQ. ' a, b, . '. = -lb + -lc - a

1.37. " a = 2U1 + u2 - 3U3, b = - 2U2 + u3, = - + 2U2 - U3' 3a - 2b + , U2, U3' '. 3 + 9U2 - 12u3

1.38. la b cl ,,: lal + Ibl + ll.

Q

. 1-18 1.39. ,

, .

1.40. .

20 .

1.41. ot . 1.42. -

.

1.43. 2 2. 1.44. 2 { . 1.45. . bi ci a. b. , () a = b,

= bi () = a + b, Ci = ; + bi () b = ka, bi = kai'

1.46. - . U2' U3 . a = - 2U2 + u3. b = u2 - U3. = 2 - U2+ 5U3 . .

1.47. - . U2. U3 = - + u2 - U3. 2 = + 2U2 - U3. 3 = 2ul + U3. . 2. 3 a = 2 - U3 . 2. 3'

. = -2 + 2 - 3 U2 = 3 - 2 + 23. u3 = 4 - 22 + 33. a = -8 + 42 - 53 1.48. - a = -e} + e2 - 2e3 b = e} - e2 + e3. . () a b. (b) lal. (c) cos 4(a. b).

(d) Pb (a). (e) Pb (a). . () -4. (b)...[6. (c) -4/(3V2). (d) -4/V3. (e) -(4/3)(e} - e2 + e3) 1.49. a = 2e} + e2 - 3e3' '. 2/...[14, /4. -3/V14 1.50. a = xe} + e2 - e3 b = 2el - xe2 + e3 .

. = 1

1.51. arlal2 - ( + )(a b) + lbl2 . . (aa - b) (ya - cSb) 1.52. - a:=-.: e} + e2 - e3 b = -e} + 2e2 - 2e3' , a. b,

. . = (2el - e2 + e3) 1.53. gl = (1/3)(2el - 2e2 + e3)' g2 = (1/3)(el + 2e2 + 2e3) g3 = (1/3)(2el + e2 - 2e3)

(el. e2' e3) (gl' g2. g3)' '. el = (1/3)(2g} + g2 + 2g3). e2 = (1/3)(-2g} + 2gi + g3)' e3 = (1/3)(gl + 2g2 - 2g3)

1.54. ! io .

1.55. ' a = e} - 2e2 + 3e3. b = 2e} - e2 - e3 = e2 + e3. () aX b. (b) b a. (c) a b = [abc]. (d) aX (b ). '. () 5e} + 7e2 + 3e3. (b) -5e} - 7e2 - 3e3. (c) 10. (d) 2e} - 2e2 - 2e3

1.56. a = e} + e2 - e3 b = -e} - 2e2 + e3' '. (I/V2)(eI + e3) 1.57. d S, a = = e} + e2 - e3

S b = -e} + e3, = e} - e2 S. '. d = IPb ,,(a)1

1.58. ( 2 3 ) det U2" U2" 2 U2" 3 . 3" 3"2 3"3

1.59. (a b) " ( d) + (b ) (a d) + ( a) " (b d) . 1.60. [(a b)(c d)(e f)] = [abd][cef] - [abc][def]. 1.61. a b d.

(a b) " ( d) = . 1.62.

U2 U3 U3 - ( , U2, U3) = _.- 2 = 3 = [23] , [23] , (, 2, Va) (, U2, U3)' 11;" Vj = j i, j = 1,2,3.

1.63. - (, u2' U3) (. 2' 3) . (, 2' 3) ( , U2, U3)'

1.64. (. . 1.27). , (, 2' 3) (wl, W2, W3) : (, U2' U3),

(, 2, 3) : (wl, W2, W3)' " : .

2

' a u () 3 u ~ . a u 3 = ku + , -00 < k < 00 (2.1)

-00 < k < 00 (2.2) (2.1) (2.2) . , k . n

, u . , u .

2.1. a = el + 2e2 u = e } - e3

= ku + a = k(el - e3) + (e l + 2e2) = (k + 1)el + 2ez - ke3 = k + 1, 2 = 2, 3 = -k.

2.2. a b , b - a . ~E, a b = k(b - a) + a

= k(b l - ) + , 2 = k(b2 - 2) + . 3 = k(b3 - a) + a3 a u v , 3 - = hu + kv + , -00 < h < 00, -00 < k < 00 (2.3) .

= hul + kvl + , 2 = hUz + kV2 + 2, 3 = hUa + kV3 + 3 (2.4) (2.3) (2.4) . , h k ,

. { , u v. , u v.

' n = hu + kv + a, - a n, (x-a)' n = (2.5)

, a n, ( - al)nl + (2 - 2)2 + (3 - a)n3 = (2.6)

21

22

. , b, c , b - a c - a

, ,

(b - a) (c - a) , . 2-1. , (2.5) , , b, c,

[( - a)(b - a)(c - a)] = (2.7) . 2-1

.2

2.3. a = e2 u = el v = -el + e3

= hu + kv + a = (h - k)el + e2 + ke3 = h - k, 2 = 1, 3 = k .

( 1 -1) [( - a)uv] = det : -1 = 3 1

. - - () a, 8,(a), - al < . ff . 2-2, S,(a),

, a . 2, 8, (a) a . , S,(a) 2 , . 2-3.

a

. 2-2 . 2-3

. -

8:(a) - 8,(a), . - al = = , S:(a) < - al < . 2.4. 1/I- a = el + 2e2 + 3e3, 8 l / lo(el + 2e2 + 3e3), = xlel + xze2 + x3e3

- al = [( - 1)2 + (2 - 2)2 + (3 - 3)2] /2 < 1/10 ( - 1)2 + (2 - 2)2 + (3 - 3)2 < 1/100 2.5. . 81/100(5) - 51 < 1/100 5 -1/100 < < 5 + 1/100. 81/ , 5 1/50.

--(

.2 23

~E 8 . t f(t), f t 8. ' ,

f(t) f tE 8 t ( f), 8 f , (8), 8 ( ).

2.6. - , b, c . '

f(t) = a - 2tb + t2c, -2 ~ t ~ 2 t -2 ~ t ~ 2. " , :

t -2 -1 1 2

f(t) a + 4b + 4c a + 2b + c a a-2b+c a - 4b + 4c

2.7. ' 2.6 a = e + 2e2, b = e2 - e3, c = el'- e3'

f f () = 1 + t 2, f2(t) = 2 - 2t, f 3(t) = 2 - t2, (eI, e2, e3)'

f(t) fl(t), f2(t), f3(t), . ' , fl(t), f2(t), f3(t) 8 f(t) = fl(t)eI + f2(t)e2 + f3(t)e3

8 (eI, e2, ~). . , = f(t). ' t ' , , . 2-4. . = f(t)

= fl(t), 2 = f2(t), 3 = f3(t) . t .

--------~-----+~----+_-------X

. 2-4 . 2-5

2.8. ' = (cos t)e} + (sin t)e2 , , = cos t, 2 = sin t > , ~ t ~ 217', ,

a. ' t ~ t "'" 2"., , . 2-5.

24 .2

, , . ,

< t < b ~ t ~ b, , ~ t < b, < t ~ b, , -00 < t < 00, ~ t < 00, -00 < t < , .

() , > , If(t)1 ~ t . ' . 2-6, , = f(t) , f(t)

, t .

1,,/2 ''1 = t

. 2-6 . 2-7

2.9. ' = tel + (tan t)e2 -/2 < t < /2 . 2-7. Ixl , t /2. ",

{ -/2 < t < /2, { -/2 + < t < /2 - > . , t

IXI = !tel + (tan t)e21 "" Itllell + Itan tllezl "" Itl + ltan tl "" = /2 - " + tan (/2 - .).

f(t) t = to, > f(t) t S.(tO) , , () to > >

If(t)1 ~ It - tol < . , f(t) ' , to . , , , f(t) to -/2 < t < '11'/2,

' .

() L t to

lim f(t) = L t .. t.

f(t) ~ L t ~ to, > > ( ) () S.(L) t S~(tO)' '

. 2-8, = f(t) ~ L t ~ to,

S.(L) S~(tO) S.(L) t S~(tO).

t.+ 8

. 2-8

/ /

f

\-- ,

" "- ......

.....

----

"-

"-, \ \

f

/ /

.. ,

.2 25

to ,

to. ' () to. , , < t < tO 2.10. " f(t) = a = . to 1im f(t) = a. , to 8.(a) f(t) == a 8.;',: t 8(t). 2..

. = ( t) = { tet + e2' t "'" tet - e2' t <

= () = t, 2 = fz(t) = , { , t"'" -1, t < . 2-9, t....,.

L 8.(L), 2 = 1 = -1. ( 8/2(0, 1) 2 = -1.) '

8.(L) > t < Itl < = f(t) 8.(L). L , . .

, t Q .., t....,. i !eI + ez .

"'. - 1

-

{ 1. ~

"'s = -1. > If(t) - LI < t S~(to). t S(t). '

If(t)j = jf(t) - L + LI ~ If(t) - LI + ILI ~ / //

= Max (, !f(to) - LI) + IL! () ~ to. :

2.2. f(t) to t -+ t o, f(t) { to.

lim () = L i , = 1,2,3'

- lim [fl(t)el + f2(t)~;+ f3(t)~] = Ltel + L2e2,+ L3e3 t ... to \

\

, f(t) = fl(t)el + f2(t)~ + f3(t)e3 L = Ltel + L;.e2 + Laea' ~ ./

lim If(t)-L! lim I(fl(t)el + fz\t)e-1. + f3(t)e3) - (Llel +L~+L3e3)1 t-+to t .... to

= lim [(() -Ll)Z + (f2(t) - L 2)2 + (f3(t) - L 3)2]l/2 ... to

=

3

26 .2

t c , : : 2.3. f(t) = fl(t)el + f2(t)e2 + f3(t)e3 t -+ to, c t fi(t), = 1,2,3, t -+ to, c

lim f(t) = (lim () el + (lim f2(t) e2 + (lim f3(t)\e3 t .... t o t .... to t .... t o t ... to 'J 2.13.

1im t)e} + (cos t)e2 + te3) = (lim t) e} + (lim cos t) ~ + (lim t) e3 = e2 - - - - 2.14. f(t) = t 2el + te2

11'm (2 + 1.) - (2) 2 + h)2el + (2 + h)e2) - (4e} + 2e2) = hm h h_O h h-O . [2 + h)2 - 4)el he2]

= 11m h +-1. h_O

f(t) -+ L t -+ to. If(t)l-+ ILI t -+ to. , f(t) = fl(t)el + f2(t)e2 + f3(t)e3 c L = Ltet + L2e2 + L3e3, lim If(t)1

t .... to lim [f~(t) + f~(t) + f~(t)1/2 t-+ to

= [( lim ft(t)2 + (lim f 2(t)2 + (lim f3(t)2]l/2 t .... to -.. to t .... t o

[L; + L~ + L~]l/2 ILI W . , If(t)1 c f(t). .. 2.11 (

to= ). V , , : : 2.4. ' f(t) -+ L }4 to, If(t)l .... ILI t -+ to.

/ . , c/ : ' lim f(t) = L, lim g(t) = c lim h(t) = , ' -. -. -.

[]

[2]

[Ha]

[IL]

[IL]

[IL]

lim (f(t) + g(t = / lim f(t) + lim g(t) = L+M -. -. t .... to

i

lim (h(t)g(t = mm h(t) lim g(t) = -. tlr. -. ' .,. , lhn (f(t)/h(t = lim f(t)/ lim h(t) = L/N.

r-'t . t .... to t .... to

lim (f(t) g(t li~'f(t). lim g(t) = LM -. t .... to "\., t .... to

lim (f(t) g(t = lim f(t) lim g(t) = L . -. -. -.

' lim f(t) = f(to), t = h() c lim h() = to, lim f(h( = f (lim h() = f(to). -. -. _. -.

2.15. lim f(t) = L, lim g(t) = , 1im h(t) = . t .... to , .... to t .... to

lim [() g(t) h(t)] = 1im (f(t) g(t) h(t = lim f(t) lim (g(t) h(t t .... to t .... 'o t ..... to t .... to

= 1im f(t) lim g(t) 1im h(t) = [LMN] -. -. -.

t

t 1

1i

\

r .2 27

f(t) to, to , > 8 > ( ) f(t)

S.(f(to t S(t) , f(t) to lim f(t) = f(to) (2.8)

t .... to

. f(t) , t = to . 2.3 f(t) fi(t), = 1,2,3, . . [] [6]

, , .

V (2.8) lim (f(t) - f(to t .... to

, h = t - to, lim (f(to + h) - f(to h .... O

2.16. " f(t) = a + bt + ct2, a, b, c . lim f(t) = lim (a + bt + ct2) = a + bto + ct~ = f(to) t .... to t ..... to f(t) { t.

t#l 2.17. " f(t) = {t; ~ : el + t3e2, f(t) { t. , tG # 1 2eI + e2, t = 1

tl~o f(t) = ~ (t; ~ : el + t3e2) t~ - 1 e + t~2 to-l -. t o = 1 ]~ f(t) = ]~ (tt2 ~ : eI + t3e2) lim t + l)et + t3e2) = 2eI + e2 = f(l) t .... t

__ {tel + e2, t "" 2.18. f(t) tel - e2' t < 2. 11 { t t = , .

f'(to) = lim .....1f(~t)_---:f~(t=o) t .... to t - to

(2.9)

, f(t) t = to f'(to) , f(t) to. ___ _

~------- , yero t = to + ~ , to ~

/' f'(to) = lim ftto + t) - f(to) At .... o \ t 2:9. " f(t) = a + bt + ct2, a, b, c { ~ .

. f(to + t) - f(to) [a + b(to ~t) + c(to + t)2j - (a + bto + ct~) f'(to) = 11m = 11m -=----=-----'\....--.....:....~-.:......---....:.....--=. At .... o t ... ' t

. b t + 2cto t + c(t)2 = 11m = lim (b + 2cto + c) = b + 2cto At ... o At ... o () to () = b + 2cfo.

(2.10)

28 .2

~E f(t) = fl(t)el + f2(t)e2 + f3(t)e3. ' 2.3

f'(to) lim f(t) - f(to) t ... to t - to

' : 2.5. f(t) = fl (t)eI + f2(t)e2 + f3(t)e3 to, fi(t), i = 1,2,3, t o,

, f(t) , f'(t) , . ' ,

, f"(t). .

' , u = f(t)

u' = ~~ = f'(t), u" = :t (~~) = ~:~ f"(t), ... 2.20. . u = (t3 + 2t)e l + (sin t)e2 + ete3 ,

u ' ~~ = :t (t3 + 2t)eI + : (sin t)e2 + :t (et)e3 = (3t2 + 2)e I + (cos t)e2 + ete3 u" :t (~~) = :t (3t2 + 2)eI + :t (cos t)e2 + : (e t)e3 = 6te l - (sin t)e2 + ete3

2.21. . = a(cos t)eI + a(sin t)e2 , . 2-10. = dx/dt = - a(sin t)el + a(cos t)e2 , , = .

2

----------~~~~~----~--------~x

/ .2-10 ! 1~K . , 2.26, :

2~/>Ci~-~f(t) to, f(t) to. /

, , j'

~ ,

.2 29

, u, v, h t , - :

[J ] . + 1 ' , ,d ( ) du dv . 1 U V dt u + v = dt + dt . d

, hu dt (hu)

u'v d dv dv -(uv) U' dt + dt u. dt

' u v d dv du dt (u ) uX dt + dt v. - ~': [J5] u = f(t) I t

t = h() .' h(I.) - l t , u = g() = f(h(B .

du du dt d dt do

2.22. - u = (cos t)et - a(sin t)e2, = (1 + 2)/2, t > .

~: ~: ~: = ~~ /~~ = (-a(sin t)et - a(cos t)e2)/[t(1 + 2)-/2} -(a!t)(l + t2)l/2sin t)eI + (cos t)ez) = h(t), de!dt =F , dt/de = l/(de/dt).

2.23. - 1t & . ~~) = ,: ( ~~) + ~~ . ~~ = u': + ~: 12 2.24. -

d [ du d2U] d ( du d2U) dt u dt dt2 = dt U' dt dt2

[ du d3u] u dt dt3

, u , \u\ = ., - U u = .,

du du du U' dt + dt u = U' dt = u du/dt. - : 2.7. u , du/dt u.

- -.

~ \ , \ - - - - 1 ij . -

. ij ~ f Cm ij Cm , - m - . - / CO, - C"'. /

30 .2

, , :

2.8. f(t) = fl(t)el + f2(t)~ + f3(t)e3 Cm , fl(t), f2(t) f3(t) Cm .

W Cm CJ j """ m. 2.25. f(t) = t-'et + ( t)e2 + t8/3e3' -00 < t < 00.

f'(t) = 3t2el + (COS t)e2 + (8/3)t5/3e3 f"(t) = 6tel - ( t)e2 + (40/9)t2/3e3 t. ' f"'(t) = 6el - (COS t)e2 + (80/27)t-1I3e3 t = , t l/3 . , f(t) C -00 < t < 00, C3. , , f C"'.

, [Jt ] [J5] : 2.9. ' f, g, h Cm , hf, f + g, f g f g Cm .

2.10. ' f Cm ' ~ t() Cm .' t(I.) , g(O) = f(t( Cm .. , Cm Cm Cm.

YLOR W f(t)

t \tQ ( Taylor) Cm .

f(t) f(to) + fYo) (t - to) -i- t

f 1

. . 2 31

2.27. ' f(t) = at4 + bt5 + c:t, , b, c: , f(t) = o(t3) t = .

lim f(t)/t3 = lim (at + bt2 + c:t3) = t-.o t ... o " f(t) = o(t2). ' f(t)". o(t") n > 3. 2.28. ' f(t) = (2 t)el + (t2 + t3)e2 + t4e3' f(t) = O(t2).

lim f(t)/t2 = lim [SinZ2

t el + (1 + t)e2 + t2e3] = el + e2 t-+O t-+O t ' , f(t)/t2 ' f(t) = O(t2). " f(t)/t .... t .... f(t) = O(t) . ' O(tZ) ({ .. f(t) , If(t)/tQI .... "" t .... > 2. 2.29. " f(t) Cm . ' Tay-lor to

f(t) C .. . m

t to f'(t ) f(to) = a"n!. ", C"'.

, , ,

. 2.30. I(t) = e- IIr t t = . '

1(0) = . I(t) C" ' -00 < t < "". " t = I(t) . 1(0) = . 1,(0) = , 1"(0) = . ... ", !(t) . 86(0). t 8(0), , !(t) t' 8(0).

, , , ,

.

32 ' .2

2.1. S (, 1, 1), Q(1, , -1), R(1, -1, ). ( :~ -~ -~)

e3 -2 -1 PQXPR det

S I~~ ~ ~:I S.

(1/v' )(3el + e2 + e3) -

2.2. (1, -1,2) 3.

2.3.

" = , a = . = - a = ke3 ij ( - 1)et + (2 + 1)e2 + (3 - 2)e3 = ke3 =1, 2=-1, x3=k+2 (-oo

r

.

! ,

.2 33

2.6. ~E (uI, U2, U3) 3, V (, 2, 3) (uI, U2, U3),

+ X2U'l + X3U3

. ' (, 2, 3) Q(YI, 2, 3)

jPQI2 = I( - )2 + I2( - )(2 - 2) + 3( - )(3 - 3) + 2(2 - )( - ) + g22(X2 - Y2)'l + g23(X'l - 2)(3 - 3)

+ 3(3 - 3)( - ) + g32(X3 - Y3)(X'l - Y'l) + g33(X3 - 3)2 iPQj2 i,j = 1,2,3 ; () = gji, (b) det () > .

IPQI2 IQPI2 = IOP-OQI2 = I-\2 = (-)'(-)

[ f ( - Yi)U;] [ f (Xj - Yj)Ui] = f f (' Uj)(Xi - )(; - Yj) \PQI2 = ~ ~ g;j(X; - )(; - Yj), gij = ' Uj, i, j = 1,2,3.

J

() = ' Uj = Uj' = gj;' i, j = 1,2.3. , 1.58

(b) ( det () = det U2 U3'

U2 ' U3) U2' u 2 u2 U3 = [23]2 > U3' U2 U3' U3

2.7. = t2el + (1- t)e2 t

-4 4 .

t

-4 16el + 5e'l -3 gel + 4e'l (16.5)

-2 4el + 3e2 -1 el + 2e2

e2 ----~~=_----------------------Xl

1 el 2 4el - e'l (16. -3) 3 gel - 2e2 4 16el - 3e'l . 213

2.8. f(t) = (1 + t 3)el + (2t - t2)ez + te3, g(t) = (1 + t'l)el + t3e'l, h(t) = 2t - 1. (a)h(2)(f(1)+g(-1, (b)jg(2)1, (c)f(a)'g(b), (d) f(t) xg(t), (e)g(2a-b), (f)f(t+t)f(to), (g) f(h(t. (,) h(2)(f(1) + g(-1 = (3)[(2el + e2 + e3) + (2el - e2)] = 12el + 3e3 (b) Ig(2)1 = 15el + 8e21 = V89 (c) f(o,) g(b) [(1 + a 3)eI + (2 - o,2)e2 + ae3] [{1 + b2)eI + b3e2J

= (1 + 0,3)(1 + b'l) + b3(2a - 0,2)

34

(e l (1 + t3) (1 + t 2)

(d) f(t) g(t) = det e2 (2t - t 2) t3 = -t4el + (t + t3)e2 + (t6 + t4 - t3 + t2 - 2t)e3 e3

(e) g(2a - b) = (1 + (2a - b)2)el + (2a - b)3e2 (!) f(to + At) - f(to) = [1 + (to + At)3]el + [2(to + At) - (to + At)2]e2

(g) f(h,(t

+ (to + At)e3 - (1 + t~)el - (2to - ~)e2 - tOe3 (3t~ At + 3to At2 + At3)el + (2 - 2to At - At2)e2 + At e3

f(2t - 1) = (1 + (2t - 1)3)el + (2(2 - 1) - (2t - 1)2)e2 + (2t - 1)e3 = (8t3 - 12t2 + 6t)el + (-4t2 + 8t - 3)e2 + (2t - 1)e3

.2

2.9. = (-1 + sin2t cos3t)el + (2 + sin2t sin3t)e2 + (-3 + cos2t)e3 '.: a = -el + 2e2 - 3e3

1.

- al = I(sin 2t cos 3t)e1 + (sin 2t sin 3t)e2 + (cos 2t)e31 = (sin2 2 cos2 3 + sin2 2 sin2 3 + cos2 2t) 1/2 { ,

2.10.

(sin2 2t + cos2 2t)I/2

== (-2 + sin t)el + (t2 + 2)e2 + (t2 - 1 + 2 sin t)e3

1

a = e2 + 2e3 = 2el + e2 - e3.

( - a) = [(-2 + sin t)el + (t2 + 1)e2 + (2 - 3 + 2 sin t)e3] [2el + e2 - e3] = , a { .

2.11. a = el - 2e2 + e3 b = 2el - 3e2 + e3. () b S3(a). (b) 8 > Sa(b)

S3(a). (c) 2 S'I(a) S'2(b) .

() Ib - al = ,,;2 < 3, b 8 3(a). (b) "" 3 - Ib - al = 3 - ,,;2. . 8 a(b), - bl < ,

- al = - b + b - al "" - bl + Ib - al < +,,;2 "" 3 - ,,;2 + V2 = 3 - al < 3 8 3(a). , 8 a(b), 8 3(a), 8 a(b) 8 3(a) (. 2-14).

::! 3-,,;2 . 2-14

(c) el = 2 "" !Ib - al = V2/2' 8'l(a) 8'2(b) { . , : 8, (a) 8, (b), - al < V2/2 - bl < V2/2. . 2

V2 = Ib-al = Ib-y+y-al "" Iy-bl + Iy-al < ,,;2/2 + ,,;2/2 . 8'l(a) 8'2(b) { .

r , :

.2 35

2.12. P(t2, -t, 2t) SI/8(1, - 1,2) t SI/Io(1).

' . t 8/0(1), It - 11 < 1/10 ( - 1)2 < 1/100. ( + 1)2 = - 1) + 2)2 = (t - 1)2 + 4( - 1) + 4

~ (t - 1)2 + 41t -11 + 4 < (1/100) + (4/10) + 4 < 5 ' t P(t2, -t, 2) (1, -1, 2)

[(t2 - 1)2 + (-t + 1)2 + (2 - 2)2)1/2 = [(t + 1)2(t -1)2 + ( -1)2 + 4(t -1)2)1/2 < [(5/100) + (1/100) + (4/100))1/2 = 1/ < 1/3

. c 8/3(1, -1, 2) c t 81/10(1).

2.13. 8 > = t2e. - te2 + 2tea SI/Ioo (e. - e + 2e8) t S6(1).

2.14.

" a = e1 - e2 + 2e3 - al 1(t2 - 1)el - (t - 1)e2 + (2t - 2)e31

~ It2-111eII + It-111e21 + 12t-211eal = it-11It+11 + It-1\ + 2\t-1\ = It-11(lt+11+3) = it-1\(\t-1+2\+3) ~ \t-11(lt-1\+5) It -11 < l' \ - al < It -1\6 < 1/100, c co It -11 < 1/600. -, c \ -1\ < 1/600, t c 86(1) = 1/600, coo c \t -1\ < 1,

- al ~ It - 1\ (\t - 1\ + 5) < It - 1\6 < (1/600)6 = 1/100 c 8 1/1oo(a), { c .

. lim [(3t2 + 1)et - t8e + e3].

t ... 2 "

lim [(3t2 + 1)e - t3e2 + e3] = (lim (3t2 + 1) e. -t-2 t_2 ( lim a) e2 + (lim 1) ea = 13e1 - 8e + ea t"'2 .. 2 2.15. ~E f(t) = ( t)e. + tea g(t) = (t2 + 1)e. + ete;

(b) lim (f(t) g(t. ..

() lim (f(t) g(t, ..

() lim (f(t) g(t = lim f(t) lim g(t) = lim t)e. + te3) lim t2 + 1)e. + ee> = (e. + e) = .. t .. o .. t .. o t .. o (b) lim (f(t) g(t = lim f(t) lim g(t) = lim t)e. + tea) lim [(t2 + 1)e. + ete2] = (e. + e> = - t_o ... ... t ... o

2.16. f(t) = B~ t e. + (cos t)e2 t = , ~ f(t) . '

" lim f(t) = lim (Sint t e. + (cos t)e) = e. + e2 .. 1 ... 0 ", f(O) = e. + e2, ~o lim f(t) = f(O), f(t) t = .

1 ... 0

2.17. f(t), g(t) h(t) to, [f(t) g(t) h(t)] to.

. ~o lim f(t) = f(to). 1im g(t) = g(to) c 2.15 ~ ... . ... .

1im h(t) = h(to). - ... .

1im [f(t) g(t) h(t)] = [f(to) g(to) h(to)] ... t.

', [f(t) g(t) h(t)] { to

36 .2

2.18. lim (t2et + (t + 1)e2) = el + 2e2 t ... t

f(t) = t2et + (t + l)e2 L = el + 2e2' " If(t) - LI = 1(t2 -l)el + (t -l)ezl ~ I2 -lllell + It -llle21

= It-llIt+ll + It-ll ~ I-I-I+2+) = I-I- I+3) It - < It - < ./4, If(t) - LI < It -14 <

", . > = min (, ./4). , It - < [ t S(), It-ll < - >

Ig(t)1 ~ < t - tol < 2 . , f(t) -+ L t -+ to , 3 > If(t) - LI < ./2 < It - tol < 3 , < It - tol < = min (, 2 , 3),

< - tol < , < It - tol < 2 < It - tol < 3 I(f g) - (L ) ~ If - Lllgl + ILllg - .( (./2)() + ILI (./2ILI) =

2.21. - u = a(cos t)el + a(sin t)e2 + bte3, , b #< . .

(b) ~71, (c) ~:~, (d) ~2t~ (.) ~~ = :t o.(cos t)et + : (sin t)e2 + :t (bt)e3 = -a(sin t)el + o.(cos t)e2 + be3 (b) ~~ = (0.2 sin2 t + 0.2 cos2 t + b2)l/2 = (0.2 + b2)1/2

du () dt'

(c) d (dU) d. d d dt2 = dt = dt (-a(Sln tel + dt a(cos t)e2 + dt be3 = -a(cos t)et - a(sin t)ez (d) :: = (0.2 cos2 t + 0.2 sin2 )/2 = 10.1

2.22. = tel + t 2e2 + t3e3 t = 1.

, dx/dt = el + 2te2 + 3t2e3' (. 2-15),

- = k dx - k dx + -00 < k < 00 dt . - dt ,

/'x~ . %-15

.2 37

2.23.

t = 1 = et + e2 + e3 dx/dt = eI + 2e2 + 3e3' t = 1 { = k(eI + 2e2 + 3e3) + (et + e2 + e3)' -00 < k < 00

= (k + l)et + (2k + l)e2 + (3k + 1)e3' -00 < k < 00

' u = (3t2 + l)et + (sin t)e2 v = (cos t)eI + ete3, () d d d dt (U' ), (b) dt (u ), (c) dt lul

d dv du () dt (u' ) u dt + dt' v 3t2 + l)el + ( t)e2)' (-( t)et + ete3) + (6teI + (cos t)e2) cos t)et + ete3)

(b)

(c)

-(3t2 + 1) t + 6 cos t d dt ( ) dv du u + dt v

det (:~ 3::/ - ~ t) + det (:~ c:: t c~ t) ~ ~ ~ ~

(sin t)eteI - (3t2 + 1)ete2 + (2 t)e3 + (cos t)eteI - 6tete2 - (cos2 t)e3 (sin t + cos t)eteI (3t2 + 6 + 1)ete2 + (sin2 t - cos2 t)e3

"

d dt \u\

(e I 3t2 + 1 cos t)

u v = det e2 sin t (sin t)eteI - (3t2 + l)ete2 - ( t cos t)e3 e3 et

d dt ( ) = [( t)et + (cos t)et]eI - [(3t2 + l)e t + 6te t]e2 - [- 2 t + cos2 t]e3

d dt (u U)1/2

(sin t + cos t)ete I (3t2 + 6t + l)et e2 + (sin2 t - cos2 t)e3

!(U'U)-t/2%t(U'U) = !(O'O)-1/22(o.~~) = (u/\u\).~~ [(3t2 + l)et + (sin t)e2]/[(3t2 + 1)2 + 2 t]1/2. (6tet + (cos t)e2)

= (18t3 + 6t + t cos t)/[(3t2 + 1)2 + 2 ]/2

2.24. " u = (sin t)eI + 2t2e2 + te3 (t > ) t = log . du/d () , (b) t.

()

(b)

du du dt de . = de = cos t)et + 4te2 + e3)(1/e). t du/de = (l/e)cos log e)el + 4(log e)e2 + e3) "

u u = (sin log B)et + 2(log2 e)e2 + (log e)e3'

du de

"

du/de = (cos log B)(I/B)et + 4(log e)(1/e)e2 + (l/e)e3 = (l/e)cos log e)et + 4(log e)e2 + e3)

= et , { de/dt = et du du dt dU/dfJ de = dt dtJ = dt dt =

38 .2

2.25. : () f(t) = , a , f'(t) = . (b) ' f(t) = ah(t), a , f'(t) = ah'(t).

() f'(t) Iim f(t + t) - () . a - a lim = lm -- = = ... ... t ...

(b) f'(t) = lim ( + t) - f(t) lim ah(t + ) - ah(t) ... '"

= Iim a 1im h(t + ) - ()

= ah'(t) '" ...

2.26 . 2.6: ' f(t) to, f(t) to.

"

lim [f(t) - ()] "' . f(t) - () lm (t - ) = "' _.

lm () - f(to) lim ( - ) "' t - to "' [f'(to)]O {: iI f(t) to

2.27. ' u v t, d du dv dt(U+V) = dt + dt W(t) = U(t) + ()

d dt (U+ )

=

dw lim W(t + ) - W(t) = ...

. U(t + ) + ( + ) - () - () :~O Iim U(t + ) - () + 1im ( + ) - () = du + dv ... '" t dt dt

d U v t, dt (U )

dv du = u dt + dt V.

" W(t) = U(t) V(t). = ddWt = 1im W(t + t) - W(t)

... = . U(t + ) V(t + t) - U(t) V(t)

~O = lim [U(t + ) (V(t + ) - V(t + (U(t + ) - U(t V(t)

...

= lim ( + ) lim v(t + ) -v(t).+ lim u(t + ) - u(t) lim v(t) ... ... t . - _

dv du = u dt + dt v

iI () ( ) 1im u(t + ) = u(t). iI V(t) t _ 1im V(t) = V(t). u v i _ .

.2 39

2.29. u = a cos kt + b sin kt, a, b { k , { d2u/dt2 = -k2u.

" du/dt = a :t cos kt + b :t sin kt = -ak sin kt + bk cos kt d2u/dt2 = - ak2 cos kt - bk2 sin kt = -k2(a cos kt + b sin kt) = -k2u

2.30. u ~~ = lul dJ~1 . " d~ ( ) = : lul2, u ~: + ~: u 2u. du dt 21ul ~~\ du u-dt

2.31. . f(t) { to, f(to + t) f(to) + f'(t)t + R(to, t) (R(to, t)/ t) -+ t -+ .

R = f(to + t) - f(to) - l'(to) t. " lim /t lim [f(to + t) - f(to) - l'(to) tj/t - _

. [f(to + t) - f(to) ] = 11m - f'(to) = _ t f'(to) - f'(to)

{ .

,d\u\ u dt

2.32. 2.31. , a t t, f(to + t) = f(to) + a t + , lim R/t = , -

- f(t) { to a = f'(to).

" . f(to + t) - () 11m = _ t

. a + lnl = _ lim a + lim / - - f(t) { l'(to) = 8.

YLOR 2.33.

(sin t)el + (t2 + 1)e2 = el + !-(72 + 4)e2 + 7e2(t - 7/2)

= a

+ !(-el + 2e2)(t - 7/2)2 + o[(t - /2)2] Taylor f(t) = (sin t)e} +

(t2 + 1)e2 t = 11'/2. " () (sin t)e} + (t2 + l)e2 (11'/2) e} + !-(11'2 + 4)e2 '() = (cos t)el + 2te2 '(11'/2) re2 f"(t) =.- (sin t)e} + 2e2 "(11'/2) -e} + 2e2 (sin t)e} + (2 + 1)e2 = e} + !-(11'2 + 4)e2 + 11'e2(t - 11'/2) + !(-el + 2e2)(t - 11'/2)2 + Iim K/(t - 11'/2)2 = .

_"./2

2.34. t = { () to(t2) = o(t3), (b) ) + ) = o(t2), (C) o(t2). o(t3) = o(t5). () lim to(t2)lt3 = lim (t/t)(o(t2)/t2) = lim o(t2)/t2 =

- t-O -

(b)

(c)

lim (o(t2) + o(t3/e2 = lim (2)/2 + lim t(o(t3)lt3) = - t_O -

lim o(t2) o(t3)/t5 = lim o(t2)lt2lim o(t3)/t3 = - ... t ... o

40 .2

2.35. ' () Cm ,

() f(to) + '~o) ( - to) + ... + ~:~)i~ol ( - to)m-t + [( - to)m] , Tay10r

('(t ) f(t) = f(to) + (t - to) +

f(m)(to) --, - (t - to)m + o[(t - to)m] = O[(t - to)m]

m.

'

[f(:~to) (t - to)m + o[(t - to)m] ]/(t - to)m ! .

2.36. ' () = o(g(t to, f(t) = O(y(t to f(t) = o(g(t, f(t)/g(t) --> t --> t o. , 2.2 f(t)/g(t) ! to. ' f(t) = O(g(t to.

2.37. ' (t) = O(Yl(t f2(t) = O(Y2(t to, fl(t) f 2(t) = O(Yl(t)Y2(t to , , O(Yl(t O(Y2(t = O(Yl(t)Y2(t.

f)/g(t) --> t --> to f2(t)/g2(t) { t = to, 2.19 (t) f2(t)

gt(t)g2(t)

2.38. ' Igl(t)1 ~ IY2(t)1 S(t), t o

(( + 0(U2(t ~ U2(t) .. . , ( (t ( ( ~ ~ UW , l ~ IU2(t)1. O(Ul(t/Ul(t) --> 0(U2(t/U2(t) --> , t -> t o, to

O(Ul(t \ UW < ./2 . . O(Ul(t + 0(U2(t > . U2(t) <

O(UI(t + 0(U2(t . U2(t) --> t ... t o ( + 0(g2(t = 0(g2(t to.

, .

.2 41

'

2.39. (I,,-I), (,,I), C(-I,-I,O). . 2 - 32 + 3 = 1

2.40. (, -1, ) = -k + 1, 2 = k + 1, 3 = 3. . - 2 = 2

2.41. 3 - 22 + 3 = 5 2 + 32 - 3 = -. . = -k + , 2 = 5k - 1, 3 = 13k (-00 < k < 00)

2.42. 1:: a . n = d, Inl ~ , = kn + a, -'" < k < "'.

2.43. a C d, c d ~ , = k(c d) + a.

2.44. (, , ), = 600. . 3xi - (2 -1)2 - (3 -1)2 =

2.45. . = (t3 + l)eI + (- t2)e2 t -4 ~ t ~ 4.

2.46. ' () = (t2 + l)e I + t3e~ g(t) = (sin t)eI - (cos t)e2' . () (a + b), (b) g(t + t), (c) f(sin ) g(t2 + 1).

. () (a2 + 2ab + b2 + l)el + (a3 + 3a2b + 3b2a + b3)e3 (b) sin ( + t)e - cos ( + t)e2 (c) (cos (t 2 + 1) sin3 + (sin (t2 + 1) sin3 t)e2 - (cos (t2 + I)( + sin2 te3

2.47. ' a = 2eI - e2 + e3 b = eI + ez + e3, b S4(a) > ( S(b) S4(a).

2.48. . lim [(t2 + l)eI + ete2 + [(t2 -l)/(t + 1)]e3]' - -

. 2el + (lfe)e2 - 2e3

2.49. t f(t) = [(t2 + l)/(t2 -1)]el + (tan t)e2 .

2.50.

2.5.

2.52.

2.53.

2.54.

2.55.

4

. t = ,-,!",n, n=0,l,2, ...

" f(t) = (t2 - l)e2 + (cos t)e3 g(t) (sin t)el + ete2' () lim (f(t) g(t, t ... o (b) 1im (f(t) g(t. '. () -1, (b) -el t ... o

, f(t), g(t) h(t) , f(t) (g(t) h(t .

, u = (t2 + l)eI - tete2 + (1og t)e3, t > , () du/dt, (b) dJ.u/dt2 . () 2tel - (t + l)e te2 + (l/t)e3' (b) 2eI - (t + 2)ete2 - (l/t2)e3

= (t2 - 2)el + (t + 3)ez + (t4 + 4t + l)e3 t = .

. = (2k -l)eI + (k + 4)e2 + (8k + 6)e3, -00 < k < 00 d

, u = (2 + t)e2 + (log t)e3 v = (sin t)el - (cos t)e2' t > , () dt (u v), d (b) dt ( v). . () (2 + t) sin t - cos t

(b) [(l/t) cos t - log t t]el + [(l/t) t + log t C t]e2 - [(2 + t) cos t + t]ea

" u = eteI + 2(sin t)e2 + (t2 + l)e3 t = 2 + 2, t "'" 2. du/de dJ.u/d2 t.

. du/de = 2(t - 2)I/2(etel + 2(cos t)e2 + 2te3) dJ.u/de2 = (4t - 6)etel + [4 cos t - (8t - 6) t]e2 + (12t - l6)e3

42 .2

2.56. :t (u. ~; - ~~. v) = ' f~ - f~ v. 2.57. Taylor () = (cos t)eI + (t2 + 2 + l)e2

t = . . (el + e2) + 2e2t - elt2/2 + t2e2

2.58. d dt [(t2 + l)eI + (l/(t + 1e2 + e3] = 2tel - (1/(t + 1)2)ez

2 59 ' ' ' ' , - " d () dv + du .. u v t, dt ' v = U dt dt v.

2.60. u du/dt = (3tZ + l)el + t3e2 - (Sin t)e3' . u = (t3 + t + Cl)el + (t4/4 + C2)e2 + (COS t + C3)e3

2.61. u d2u/dt2 at2 + bt + , a, b, c - . . u = at4 + !bt3 + !2 + Clt + Cz

2.62. u du/dt = , u .

2.63. (tan2 t)e l + (2t3 + t4)e2 = O(t2) t = .

2.64. ' f(t) t = to

2.65.

f'(to) = , f"(to) = , f(n+ 1> (to) = f(t) f(to).

= {-(/)' t "'" ' () f

i

3

Ko;l ()

= X(t), t (3.1) '

() x(t) () x'(t) #- t . ( . ' , ' .) ' t .

' 3, = x(t) - = xI(t), 2 = X2(t), 3 = X3(t), tEI (3.2) = x(t) . = x(t) , = XI(t)

CI t ; (t) .

3.1. = (t+ 1)eI + (t2 +3)e2, -00 < t < 00, , ' = eI + 2te2 #- t. . 3-1.

(-3, .. ) ,=0

--------~----------------X

.3-1 . 3-2

3.2. r = 2 cos ( - 1, ~ ( "" 2'/1", . 3-2. = r COS (, 2 = (. ' ' ,

= (cos 8)(2 cos 8 - 1), 2 = ( (1)(2 COB ( - 1), ~ 8 ~ 2'/1" = (COB 8)(2 COS 8 - l)ei + ( 8)(2 cos 8 - l)e2

43

44 .3

' ,

[-4 sin cos + sin o]et + [2 cos2 - 2 sin2 - cos o]e2 Ix'l = 5 - 4 cos ~ , ~ .

= x(t) , t l # t 2 x(t t ) = x(t2). . 3.7 :

3.1. ' = x(t) , to to x(t) -. 3.3. = a(cos o)et + a(sin o)e2' ~ , (-00 < < 00) , ll, dx/do = -a(sin o)el + a(cos o)e2

Idx/dol = I-a(sin o)el + a(cos o)e21 = ll ~ ,

cos ( + 2)e l + sin ( + 2)e2 = a(cos Do)e l + a(sin DO)e2 , = a(cos o)el + a(sin o)e2 .. - t".. < < + 1' - ( ).

3.4. '

. 3-3, t t = dxl/dt = dXz/dt = . "

. t = . , , > .

> , 1/2 < , t l = -(1/2) t2 = +(1/2). , - < t l < t2 < ' XI(t l ) = 1/422 = XI(t2)

X2(t l ) = = (1/422) sin 2 = X2(t2) - < t < .

t:"' { , , t2 sin (1/ t), t >

-00 < t < 00

2

. 3-3

13

t = t() . ~f..--*'Y1 tl !1u. (i) Cl . () dt/d ~ .. , t = t() dt/d dt/d # .

V dt/d > t = t() C, dt/d < Cl. '

(. 3.13) : 3.2. ' t = t() .'

(i) t = t() - . 1t = t(I.) (ii) = (t) I t ,

. 3.5.

() ' t = (b - ) + , :"' :"' 1, < b, ll :"' :"' 1 :"' t :"' b. ' = (t - a)/(b - ) !l

:"' t :"' b :"' :"' 1.

1

;

.3 45

(b) ' t == tan (11'/2), "'" < 1, "'" < 1 "'" t < "'. == (2/) Tan- l t "'" t < 00

"" < 1.

= x(t), t I t , = *(), .' t = () . (ii) (()) = *() 3.14

. . ( , .)

W = x(t) C ' . ' C = x(t) ... = x(t)>>.

. , ' .

' , , .

3.6. == t + 1, -1 "'" t "'" 211' -1, -'

(cos )(2 cos -1)e} + (sin )(2 cos - l)ez 3.2. '

[cos (t + 1)] [2 cos (t + 1) - 1]el + [sin ( + 1)] [2 cos ( + 1) - 1]e2' -1 "'" t "'" 211' - 1

.. t -1 "'" t "'" 2 - 1, = t + 1 "'" "" 2 .. , , . 3-4(a).

= -t, -211' "" t "'" , = (cos t)(2 cos t - 1)et - (sin )(2 cos t - l)ez,

-211' "'" t "'" ' t -2". "'" t "'" , = -t "'" "" 211 , , . 3-4(b). ,

' . ,

{

t, "" t "'" 11/3 -t + 211, 11'/3 < t < 511'/3

t, 5,,/3 "'" t "'" 211' e(t)

= {cos e(t)] (2 cos e(t) - 1)el + [sin e(t)](2 cos e(t) - 1)e2' , [. 3-4(c)J. , . ,

, .

() (b) . 3-4

(c)

46 .3

3.7. 8 ,

a(cos t)et + a(sin t)e2 + bte3, , b >F , -ao < t < ao = cos t, 2 = sin t, 3 = bt, , b >F , -ao < t < ao . 3-5. " () ll = a cos t, 2 = a sin t, -ao < 3 < ao. " 3 = bt 3' 80 t 217", 2 , 3 (b > ) (b < ) 217"lbl, .

3.8. "'" t "'" 1 Q: "'" "'" 1, "'" 2 "'" 1 2. . .. 2- . , Peano,

: Q , Qo, Qt, Q2' Q3. Q; Qio, Qil' Qi2' Qi3

' , ... " , , , ,

. 3-6. .

. 3-5

: (, b > )

2b

~ ------+2

Qll

Qlo

Q03

Qoo

Q2:r -

Ql21 Q22 .. L.J

Ql3 Q20 Q23

Q3lr io--

Q021 Q30 f

Qol Q32 Q33

. 3-6

to "" t "'" 1 a2 3

= 10 + 102 + 103 +

, to a2 a3

= - + - + -- + ... 4 42 43

""" "'" 3. ( ,

1 2

1 1 3 3 3 1, .. '4 + 42 + 43 + 44 + = 4" + 42-)

to = ~ a;l4i - Q -

QQ t ' QQlQ2' QQlZQ3' " ( Q),

Q .

, . , S.(Po) (. 3-7), . 3-7

QQl Q2" .,,' , , ( ) S.(Po). ' t to

i

..

.3

-1 + -"--

4" . S.{P ). , .

47

' -, . , -.

!, = X(t), t , , t l -=F t2 X(tl) -=F X(t2).

.

, = X(t), t , ~ t ~ b. ( ' .) x(a) x(b) . '

. ', = x(t), t , , = x*(t), t 1*, 1* x*(t) x(t) 1*,

. 3.9. 3.2 ,

(cos )(2 cos - 1) 2 = (sin )(2 cos - 1) ( )

=: =: 2".. ' . . =: =:",

. 3-8.

-----~-~---_,----X

. 3-8

' = x(t) = *() . ' dt/d > , t =>x(t) = *() ' . ' dt/d < , t

= x(t) = *() . .

, .

= x(t) C . 3-9. to

= xl(to)el + X2(tO)e2 + ke3, -00 < k < 00

= xl(to), 2 = X2(tO), 3 = k, -00 < k < 00 2 x(to)

. = xl(t), 2 = X2(t), 3 = k, -00 < k < 00 (3.3) . 3-9

48 . 3

(), 2 C .

. (3.3) 2, 3 = , () = x(t) 2. = x(t)

= xl(t), 3 = = x(t) 23 3

Xl(t), 2 = , 3 X3(t)

X3(t)

3.10. = t, 2 = t2, 3 = t3, -00 < t < 00, 2 = t, 2 = t2, 3 = . 3

= t, 2 = , 3 = t3 , = t, 3 = t3, -00 < 2 < 00 . 3-10.

. 3-10

. (, 2, 3)

F l (Xl,X2,X3) = F 2 (xl,X2,X3) = (3.4) . (, 2, 3)

(iJFlIiJXl iJFlIiJX2) det ~ iJFz/iJxl iJFz/iJX2

(3.4) 2,

= (3), 3 = 3 3. (3.4).

3.11. 2 - : = 3 - X~ = = t3, 2 = t2, 3 = t ( = t, 2 = , 3 = ). ,

3 ~ ( ), 2,

3 = t

' 3 = , 2 = : = . ' = t, 2 = , 3 = . (, , ) .

r

.3 49

Cm = X(t) Cm, m ~ 1, = x(t) Cm . ',

t = t( ) . cm, t( ) Cm .. , Cm ' Cm,

Cm. ~E, Cm Cm, Cm.

= X(t) Cm C; j ~ m . . ' C; j < m, Cm = X(t) cm,

C; j < m = X(t) C; Cm. 3.12. W(t) = a(cos t)et + a(sin t)e2 + bte3' -00 < t < 00, . ", = w(t)

, , ' .

3.13. .

t < t =

t > ! C'" (. 2.30, . 31) C'" C''' . 3-1 . t < 3, t >

2'

.3-11

. ' C, ' , = x(t), ~ t ~ b. to < t l < ... < tn b

~ t ~ b. 3 = X(tO), = X(tl), Xn = X(t n ) , . 3-12. - Xi I - -l.

n n

s(P) - - IX(ti ) x(ti-t)1 (3.5) = =l

. , ', ,

. 3-12. ' , ' ', s(P) ~ S(P'). ~E

/-~~----~ . 3-12

50 .3

C C? . " .., , S

s(P) . S (supremum), .

uEva S , "'" S. S. , S, L

"'" L . , S ,

, s , L , L ~ . W C ( ). , = x(t) = *() C I t . , t = ()

-. 00 < < ... < 0n . to < t l < ... < t .. I t , t n < t n - l < . " < to ( ), = t(Oi), = , 1, ... , , - = (), = , 1, ... , , , . , S

S, C. 3.14. = tel + t2e2, "" t "" , .

0= to < t l < ... < tn = . !

n

S(P) ~ (tiel + t~e2) - (ti-tet + t~-le2)1 =

~ (ti - ti-t)et + (t~ - t~-l)e21

"" 3 ~ (t; - t t - t ) = 3

"" t i - l < ti "" , + t i - t + ti "" 3 ~ (t; - t t - t ) = tn - to = . , s(P) 3. , s(P). 3.15.

= t

2 = {tocos (l/t) 0< t"" t = "" t "" (. 3-13) . ,

, /( - ), ... , /2"., /, ,

s(P) = ( ~ ) el + ( ~ ) [cos ( - )]e21 + [( ~ 2) - ( ~ )J el + [( ~ 2) cos ( - 2) - ( ~ ) cos ( - )J e2 1 ~\

, ;.,,1\ . . ~,p ~{'~ ,

j'" . . - /.;

"-~/

r

r

\.

.3

"'" -2 1 [ 1 1 ] [ 1 1 ] s(P) .. ~ ". - ( + 1)". el +. ". cos ". - n + 1 cos ( + 1)". e2

"'" Ni21[~cosn". - +IICOS(n+l)".]e21 .. = ". n

, , lae + be21 "'" Ibe21' . , { ,

-2 : _1_ . W, s(P) .. = n + 1 , . ! .

3.23 3.24 :

51

= t

.3-13

3.3. ' { . ' = x(t), 6 t 6 b, { ,

s (3.6)

3.16. = ( cos t)el + ( t)e2 + bte3, "" t "" 2"., !

8 = 2 a2 sin2 t + a2 cos2 t + b2 dt = 2 (2 + b2)1/2 dt = 2".(a2 + b2)1/2

' = x(t) .

s s(t) dx dt J to dt (3.7) ' t:=: to, s:=: (3.7) x(to) x(t). ' t < t o, s < (3.7)

x(to) x(t). ' (3.7)

ds dt = = I~~I , s = s(t) { [. ', s(t) { Cm I,dv X(t) { Cm ' . ~E s

, X(t) (3.7), t = t(S).

~ { , to

( 8 = ) , . ..

~-----~ ..... ,.

52 .3

s(t) = - f.t [dXI dt to dt

", = X(S) . 11 , dx/ds = 1. s

(. 3.19 3.20): ~ 3.4. ' = x(s) C, :

() C () X(S2) IS2 - l (ii) , = X*(S*) C, s = S* + .

() ' = X*(t) C = X(S), ds/dt = Idx/dtl. ' '

, ds/dt = -Idx/dtl. " , s = s(t) (3.7), = x(t(s ,

I~:I / dx / dt/ dt ds /~~I/I~;I /~~//I~~I 1 3.17.

= ( cos t)eI + ( sin t)e2 + bte3

s = J t dx dt = f (2 + b/2 dt = dt t = (2 + b2j-lI2S ,

, s . '

dx d2x ,_ dx ,,_ d2x ds , ds2 , - dt' - dt2'

3.1. = tel + (t2 + 1)e2 + (t -1)3e3

t 2 3.

dx/dt = el + 2te2 + 3(t -1)2e3 { Idx/dtl = [1 + 4t2 + 9(t - 1)4)112 #- t. { t. . 2 {

= t, 2 = t2 + 1, 3 = 2 = X~ + 1, 3 = .. 3 { = t, 3 = (t - 1)3, 2 = 3 = ( - 1)3, 2 = . . 2 = X~ + 1 3 = ( - 1)3.

3.2. = (1 + cos ), 2 = , 3 = 2 (/2), -2"'" "'" 2, 2 ( -1)2 + ; = 1.

r

r . 3

(/ = - , dX21do = cos , dX3Ide = cos (/2)

[1 + cos2 (/2)] 112 # .

; + ; + ; (1 + COS )2 + Sin2 + 4 sin2 (/2) (1 + COS )2 + sin2 + 2(1 - cos ) 4 ( -1)2 + ; = cos2 + sin2 = 1,

2 () ( -1)2 + ; = 1 . . 3-14.

53

. 3-14

3.3. ' { r = 2 sin tan , -7

54

, e l ,

= 40 + e - 11" = e ~ + e - 11"

= (cos p)el + ( p)ez

ro + r --e-11"

r

.3

= = + = [ (ro ~ r) cos e .

( ro + r \] [ (ro + r )]

- r cos --r-e) el + (ro+ r) sin e - r sin -r-e ez

3.5. ' = 3 = 1,

= 2 .. /3

= 4 cos - cos 4, 2 = 4 sin - sin 4 ( ) .

" dxI/d8 = -4 sin 8 + 4 sin 48 = sin e = sin 4e e = 2n"./3, (2 + 1)"./5, n = , 1, ....

' dX2/de = 4 cos e - 4 cos 48 = cos 8 = cos 4e e = 2"./3,2"./5, n = , 1, ....

. , e = 2n"./3, n = , 1, .. " 2"..

}-----iE--- =

. 317

3,6. ( ) 2+ 2 1 . 2 = + 2 + 3 = 1.

' , = cos e 2 = , ~ e ~ 2"., 3 = 1 - - 2 = 1 - cos e - sin e. "

= cos 8el + ee2 + (1 - cos 8 - sin e)e3, ~ e ~ 2". { .

3.7. ' g(t) t = to g(to) :/= , 8 > g(t):/=O t S(t). 3.1. , = x(t) , to to x(t) -.

= !lg(to)l' ' g(t) to, ~ 8 > /g(t) - g(to)/ !lg(to)l ' g(to) . , ~ g(t).p t S(t).

, = x(t) . to , , (to). () { to. , , 8 > (t) . t S(t). ' x(t) - S(t) , t l , t 2 =F t z x(t l ) = x(t2), xt(tt ) = xt(tz). ' , ,

) - xl(t2) , = t _ t = (t'), t l < t' < t 2 2 , (t) =F ' S(t). " .

r .3 55

3.8. . = X(t) { ; (to) -F , to, = x(t) 2 = Ft(xt), 3 = F 2(xl).

x~ (to) #- S(t), = Xl(t) - t = () . . t 2 = X2(t), 3 = X3(t)

2 = X2(t(Xl, 3 = X3(t(Xl 2 = F t (), 3 = F 2(x t ),

3.9. t = (2 + 1) {

< < 00 < < 00 < t < 1. ' dt/de = 2/(2 + 1)2 dt/de #- < fI < 00. ) < < 00. ' 2/{2 + 1)l= = lim 2/{2 + 1) = 1, < < 00 < t < 1. -.",

3.10. t = Tan- l (/4) = cos , 2 = , - ~ := .

1 t2 t4 t4 - 6t2 + 1

cos = cos4 (/4) - 6 cos2 (/4) sin2 (/4) + sin4 (/4) = (t2 + 1)2 - 6 (t2 + 1)2 + (t2 + 1)2 = (t2 + 1)2

sin = 4(sin (/4) COS3 (/4) - 3 (/4) cos (/4 = 4 (t2 11)2 -4(t2 ~ 1)2 = 4/t~\-1~; _ (t4 - 6t2 + 1) - - (t2 + 1)2 _ - 4(1- t2) -1"""'- t """'_ 1, ' .. . -~ . 2 - {t2 + 1)2 '

3.11. t = 2

3.12.

= 2 2 , 2 = 2 2 tan , -/2 < < 7/2 2 t = (. . 3.3.).

" = 2{!t)2 = it2, 2 = it2 tan (Sin- l it) = !t3{4 - 2)-/2. (4 - t 2)-l/2 ! + -ht2 + O(t2). = it2, 2 = it8 + -ht5 + O(t5).

( ) t = 2 = ~X3/2 + o(x~). :

:= t < 1. , .

t = t() (i) ~ t ~ 1, () < t < 1, () -

" ~ 3.5 45, t = ( - a)/(b - ,) , a "" "" b "" t "" 1, a < e < b

< t < 1 , "" < b "" t < 1. . t = -( - a)/(b -) + 1 a < "" b "" t < 1. . . = Tan - 8 -00 < 8 < 00 -11"/2 < < 11"/2 t = ( + !11")/11" -11'/2 < < 11'/2 < t < 1. , t = {11"/2 + Tan- l 8)/11" -00 < 8 < 00 0< t < 1. .

= Tan- l 8 "" 8 < '" Tan- l a "" < 11'/2 t = ( - Tan- l )/(11"/2 - Tan- l ) Tan- l "" < 11"/2 "" t < 1. , - t = Tan/;l8 ; Ta~-l a "" 8 < 00 "" t < 1. -

11" - an .

" . ", .

56 .3

3.13. 3.2: ' t = t() .' 1-1 = ()

I t = t(I.). dt/do dt/do "= , dt/do > dt/do < 18' dt/do > ' t(o) . ,

() ~ () < 2 , ~ ) - t(02) t'(o')

- , dt/do > ' 18' t(O) , - , , o(t). , () ,

o(t) . (' .) ' ()

do lim 1/ lim == 1/ dt dt "' t ... d , dt/d . , .

3.14. . = x(t) I t { = *() .' t = t() ( ) = I t x(t(B = *(). .

== x(t) , t == . , = x(t) = *() t == t(), == *() ! == x(t) == (t), (l) == l *((t == (t((t) == x(t). , == x(t) ! = *() t == t() == *() == **() == (). t == t((. ' ddt _ ddt dd !

. dt/d "= l. , t == t(( l. t((/ == t(Ie) == l (t(() = *(( == **(). , = x(t)

= **(), .

3.15. ' = 3(cosh 2t)el + 3(sinh 2t)ez + 6te3, ~ t ~ 71'.

"

8 == ~ ~~ dt = ~ 16 sinh 2tel + 6 cosh 2tez + 6e31 dt = f" 6 [sinh2 2t + cosh2 2t + 1]1/2 dt = i 1T 6[2 cosh2 2]/ dt == 1'11' 6V2 cosh 2 dt

3V2 sinh2

3.16. (. 3.4)

= (ro + ) cos - r cos (ro : r ) = (ro + ) sin - r sin (ro : r ) .

" ~ [(:)2 + (~:)2T/2 do 8 ~8(ro+r{(-sino + sin(ro;ro)X + (coso -cos(ro;ro)5I/2dO (ro + ) 58 [2 - 2 cos (/)]/2 do 2(ro + ) 58 sin (/2) do

(ro + ) 18 (ro+r)r - 4 cos (roo/2r) = 4 [1- cos (/2)]

ro ro

r -)

.3 57

3.17. = (e t COS t)el + (e t sin t)e2 + etes, -00 < t < 00

8 - ~ t 1 ~~ dt = ~ (e t cos t - et sin t)et + ( sin t + e t cos t)e2 + eIe31dt = ft [e2t(-2 cos t sin t + 1) + e2t(2 cos t sin t + 1) + 2!2 dt = V3 dt = V3 ( -1)

t t = log (s/V3 + 1), -V3 < 8 < 00. _ r;; = (81V3 + 1)(cos log (s/V3 + 1)el + sin log (8/V 3 + 1)e2 + e3)

3.18. = !(s + VS2 + 1 )et + !(s + y'S2 + 1 )-te2 + !V2 (log (s + y'S2 + 1 es , dx/ds = 1.

V u = 8 + ys2 + 1. = !uet+ !u- le2+!V2(logu)e3

(!el - !U-2e2 + !V2U-le3) (1 +~) V82 + 1

~;

dx dx du ds du ds

~: 11 ~: = 8 + 82 + 1 ~(1 + u-4 + 2u-2)1/2----2 82 + 1 82 + 8Vs2+ + 1

(8 + S2 + 1) 82 + 1 !.(l +u-2) __ u_ = 1. u 2 + 1 2 82+1 2 UY82 + 1

, ; Idx/dsl = 1, s .

= 1

3.19. . 3.4(): ' = x(S) 1s, = x(s) X(St) X(S2} IS2 - sl\'

, 8 :! 82' { f8'1

~: d8 = fS' 1 dsl=.82 -81 = 51 81

182 - 811' . 8 > 82' ~o { f81

: ds = f8. d8 = 82 82 8 - 82 = 82 - 811'

3.20. 3.4(ii): ' = X(S) = X*(S*) , s = S* + .

" 8 = 8(S*). dx dxds ,ldXlldXlldSI d8* = ds ds* d8* = ds d8*' ~: = :; = 1. \ :!ss* = 1 1'j :!s8* = 1 8 = 8* + .

3.21. = t 2et + sin t e2, """ t """ -/2, .

5

= to < t1 < t2 < ... < t n = /2

n

8() = ~ ;-- =1

:! ~ [(~ - t[-I) lell + I - t i - 11\e211

58

W - , -

8() "'" ~ [(tj - tj_t)(t j + t j - t ) + Icosejl(tj - - )] = ~ (tj - tj-t)[t, + t'-l + Icose,IJ

Icos l! "'" 1 (t; + t l - t ) "'" ( "'" t l - t < t l "'" /2), 8() "'" ( + 1) ~ (t j - tl-t) = (/2)( + 1)

8() , .

.3

3.22. . = f(t), ~ t ~ b, , > > = to < t I < ... < t n = b

() t; - t;-t < , = 1, .. . ,n () 18 - 8()1 < 8 8() = f(t) .

' s s(P), = t~ < t~ < . .. < t~ = b ' 8(') > 8 -.. , - , 8() "'" 8 -. , 8 -.

8() - 8, . W (), = t o < < '" < t n = b (tj - t l - t ) < . ' ',

' , 8() "'" 8(') "" 8 18 - 8()1

() 1/:(t) - I:(t') < 9(6 ~ )' = 1,2,3 It - t'! < . ', , 2 Itf - tt-tl < 8 -

() fb If'(t)1 dt - ~ If'(ei)l(tj - -) < ./3 tI-l "" "'" t j -

1

. 3 59

8 = min(8 i .82). 3.22 = to < t l < ... < tn = b ., ( - - ) <

(iii) IS - s(P) < ./3

= \ s - ~b If'(t)1 dt \ ~ is - s(P)1 + \8() _ ~b If'(t)1 dt \ ~ i + \ :t If(ti) - f(ti-t)1 - ~b If'(t)1 dt \ ~ i + \ ~ I(I(ti ) - I(ti-tel + (!2(t i ) - !2(ti - t e2 + (f3(t;) - !3(ti - te31 - f

a

b

If'(t)1 dt \

~ ./3 + \~I!~(t{)el + !~(t;')e2+/3(t:")e31(ti-t;-l) - ~blf'(t)ldtI : If'(t;>1 (t; - t;-t), f

~ ./3 + \ ~ f'(ti)(t; - t i - t ) - ~b f'(t) dt \ + \ ~ [1/; (ti)et + 1;(t;')e2 + 1~(ti")e31 If'(ti)l] (t; - t,-t) \

() Ill - Ibll ~ l + bl,

, (), " ....:' 1< 3+3+3(b-a)f(t;-ti - t ) = ,

,

= 18 - ~b If'(t)1 dt\ fb If'(t)1 dt

8

" 3.25.

= tel + (t2 + 2)e2 + (t3 + t)e3 t 3 2'

3.26. r = ~ + c, .,. , c.,. , -.". ~ IJ ~ .".. COS 11 . '. = + C COS 11, 2 = tan 11 + C 11

3.27. x~ = : = 1- , . : : + x~ = 1.

'. = cos2 11, 2 = 11, 3 = COS '11, ~ 11 ~ 211"

60

3.28. C, C

t Co, . 3-18. ' C Co

(, ), .

'. = (TO-T)COSO+TCOS(rO~rO) 2 = ( - ) sin - sin (ro ~ r )

3.29. ' = 5 r 2,

= 3 cos + 2 cos 30/2, 2 = 3 sin - 2 sin 30/2 .

. = (4/5)n"., n = , ::1, ...

.3

.3-18

3.30. = 3t5 + 10t3 + 15t + 1 t.

3.31. , < t"'" 2 -00 < "'" .

3.32. = et(cos t)el + et(sin t)e2 + ete3' "'" t "'" . . 3( -1)

3.33. (. 3.28)

(rO-T)COSO+TCOS(rO~rO), 2 (TO-r)SinO-TSin(rO~rO), > 4r(ro - ) . '. 8 [1 - cos (/2)]

ro

,

3.34. tel + (sin t)e2 + ete3' - 00 < t < 00 (log t)el + sin (log t) e2 + te3, O

r ,

4

" , 11 . , , ,

--, ...

. , , .

ill ~E = X(S) C. j!froUf;vlJ C

X(S), dx/ds = X(S). ,

( ) . x(s + S) - x(s) xs = 1m ..... s

" x(s + S) - x(s) { " ' ' s - C, . 4-1. x(s) ,

Idx/dsl = !i! = 1 . = x(s*) 52 s = s* + .

dx dx ds ds* ds ds*

. 4-1

C, 3.4 -

= dx ds , dx/ds* 11 dx/ds, = x(s*). i(s) . "

. 4-1, x(s) s . i(s) () = x{s) x(s) t(s) 11 x(s) t. 4.1. = a(cos t)e1 + a(sin t)e2 + btea, > , b.p ,

~: = -a(sin t)e1 + a{cos t)e2 + be3 ~: = (a2 + b2)1/2 .

t = : = :;: = :/~ = ~~/I~~I = (a2+b2)-l/2(-a(sint)el+a(cost)e2+bea) ds/dt = !dx/dt! (. 3.4). ~o t e = Cos-l (t e3) = Cos-l b(a2 + b2)-1/2 :1:3'

. ds/dt = Idx/dtl. , .

61

62 .4

. = X(t) = X(S),

C

dx dx ds ' = dt = ds dt = ds/dt = Idxldtl. ~E, , X'(t) ' t(S)

, t = ' ! (4.1)

C

C (. 4-2). (2. ) 21, 1) .. J.Y~WHQIIV

= x(to) c/ + kto, -00 < k < 00 t = t(to) -

. - 4-2

C . ' (2.5) 2 , .. Lx,

, - ) t = , .. , = X(t). C

= x+kt,

-00 < k < 00

(-) t =

(4..2)

(4..3) ' t

= + kx', -00 < k < 00 . (-)' ' = 4.2. = te l + t2e2 + t3ea t = 1

= (1) + kx'(1) ij '= (1 + k)et + (1 + 2k)e2 + (1 + 8k)e3. -co < k < co t = 1

(-(1'(1) = ij (Nl-1)+(N2-1)2+(Ya-1)8 =

= X(S) Cm, m ~ 2.

t = t(s) = x(s) { r. dt/ds = (S) x(s)

1

1 \

, . 4 63

t C, i . , = x(s*) C t* = dx/ds*, s = s* + .

dt* _ ~(dX) _ ~(+dX) _ +~(dX)~ _ (+l)2~(dX) = dt ds* - ds* ds* - ds* - ds - - ds ds ds* - - ds ds ds

, t . t(s) C X(S) k(s)

k. t -, 2.7 29

.

k = t t . ' ,

, . 4-3. '

\K(S)\ \k(s)\ (4 .. C x(s).

1

p(s) = IK(S)I X(S).

1 Ik(s)1

. 4-3

(4.5)

' C, k = , . W, ' ll K~ .

4.9 73 . , , ..

, , , . vvJ\-'}.,S

4.3. t = a(COS t)et + a(sin t)e2' f > , f ~~ = -a(sin t)et + a(COS t)e2' \ : =

t ~~/I ~; = -(Sin t)et + (COS t)e2 dt k = = ds

" k f . = Ikl = / = / = . ", ,

.

4.4. = a(COS t)el + a(sin t)e2 + bte3' > , b =F ,

dx dt -a(sin t)et + a(COS t)e2 + be3' f ddXt -- (2 + b/2 t ~~ 11 ~; \ = (2 + b2) -1/2( -( t)et + (COS t)e2 + be3)

64

k = t = ~:/I ~: (a2 + b2)-1/2(-a(cos t)e1

- (sin t)e~/(a2 + b2)1/2

k i :1:1:1:2 :1:3, . 4-4. {

ll = Ikl = a/(

.4 65

n(s), C k(s) == K(S) n(s) (4.6)

, n(s) k(s), K(S) == Ik(s)l. ' n(s) k(s), K(S)== -lk(s)l. '

, k(s) == , K(S) == . . K(S) (4.6) ()

C X(S). , n(s) , K(S) . ,

ll == Ikl. , , . ' (4.6) n(s) n n == 1n\2 == 1,

K(S) == k(s)' n(s) (4.7) (4.6) (4.7). 4.5. . 4-5,

== tel + !t3e2'

~~ el + t2e2, ~~ == (1 + tJ2, t == ~~/ ~~ == (1 + t4)-l/2(el + t2e2) k == == ~: == ~:/ ~~ == -2(1 + t4)-2(t2el - e2) . k . 4-5().

2

() k (b) Ilk == k/!k! .4-5

'" {-Uk (c) n == e2 uk

t < t=O t>

t = { k == . Uk { , . 4-5(b). , Uk' t t, {

Uk' t t, {

lim ~ -- Ikl

-(t2et - e2) = ~~ (1 + t4)1/2

. t2et - e2 = }~ (1 + t;4)1/2

t tI t>O t <

'

= e2

66 .4

{

-k/1kl e2

k/lkl

t < } t = = >

0(8) , . 4-5(c). (4.7)

= k = [-2t(1 + t')-2(t2eI - e2)] [-(1 + t4)-1!2(t2eI - e2)] 4.6. C'" (

. 3.13, . 49)

t < t =

t>

' . 4-6, 3 t < 2 t > . k 3 t < 2 t > . ", ! ,

, ! t = , k 3 2'

2t(1 + t4)-3/2

"""".-.------- 2

4-6

' , C'" . ' , .

, 4.15 75 : 4.3. , , , .

C (. 4-7) C .

= + kn, -00 < k < 00 (4.8) C .

2.3 22 il

[( - x)tn] = (4.9) . 4-7 t = t = n, =F ,

[(-)] = (4.10) ,

r

.4 67

. ", ,

. . - .

.i. . 4.7. = (cos t)et + (sin t)e2 + te3' "

' = (- sin t)et + (cos t)e2 + e3, Ix'l = V2

. k "" t,

'/l = (1/V2)(- sin t)et + (cos t)e2 + e3) k = i = t'/lx'l = -(-!)(cos t)e t + (sln t)e2)

n = k/lkl = -cos t)el + (sin t)e2) . t = ... /2

= (/2) + kn(/2) ij -

68 .4

co

.4-9

~E, C :

, : :

:

: : , :

+ kt + kn

+ kb

( - ). t = ( -). n ( -). b =

4.8. 4.4

= a(cos t)e l + a(sin t)e2 + bte3' > , b "c t = (2 + b2)-l/2(-a(sin t)et + a(cos t)e2 + be3)

k - 2 : b2 cos t)et + (sin t)e2)' = I~I = -cos t)et + (sin t)e2)

(

e l -(2 + b2)-l/2 sin t b = t n = det e2 (2 + b2)-l/2 cos t

e3 b(a2 + b2)-l/2

- cos t) - sin t

= (2 + b2)-l/2(b(sin t)et - b(cos t)e2 + ae3)

t = to = ( to) + kb(to) = ( cos to + kb(a2 + b2)-l/2 sin to)el + ( sin to - kb(a2 + b2)-l/2 cos tO)e2

+ (bto + ak(a2 + b2)-l/2)e3' -00 < k < 00 vH, = k(a2 + b2)-l/2,

= ( cos to + eb sin to)et + ( sin to - eb cos tO)e2 + (bto + ae)e3' t = to [

( - x(to . n(to) = ( - cos to)(- cos to) + (2 - sin to)(- sin to)

cos to + 2 sin to = [ 3.

-

r

. 4 69

= x(s) Cm, m ~ 3, n Cm', m' ~ 1. ' b = t n,

b(s) = t(s) n(s) + t(8) () = K(s)[n(s) n(s)] + t(s) () = t(s) () (4..13) (4.6) aX a = a. ' n(s) : () n(s), . () t(s) b(s), ~~ ~ ~~+~~

' (4. 3), .

b(s) t(s) [.(S) t(s) + '1'(S) b(s)] '1'(S)[t(S) b(s)1 .

b(s) (t(s), n(s), b(s) ~) b(. , b #

db b = d8

. ,

db/ldXI . dt = (a2 + b2) -t(b(cos t)et + b(sin t)e2)

= -b' n = -(2 + b2)-l(b(cos t)el + b(sin t)e2) - cos t)et - (sin t)e2) = b/(a2 + 02) ~ , b > ( > ), , . +O(a).

b < ( < ), , . 4-IO(b). , .

70 .4

() ~K, ". > (b) ~K, ". < .4-10

, = X(S), _ d ds(x'bo) x.bo t'bo

, t bo, d ds(x.bo) , ,

' bo . (4.16) .!Jjl x~ X(S) l~

7JJ2J: "1}0 - . : = X(S) - , . 4-11. ' ,

:

.4-11

2 4.4. Cm, m ~ 3, ( ), Cl, , .

~ , cm, m ~ 3, CI. , t, b Cl.

, 4.19 77, . 4.5. = X(t) '>'= . ['""']

= /' '

~ C, ,

1, . 4-12 . ( ) t.

, = x(s) C, = t(S) = X(S) .

, s = t(S),

~~ = ~: /t/ / . 4-12

r . 4 71

. , = t(S) , = x(s) \1(\ == 1.

n [ 2 = n(s)] b [ 3 = b(s)J. - 4.10. 4.8 4.9

= a(cos t)el + a( t)e2 + bte3 > , b ~ t = (a2 + b2)-ll2(-a(sin t)el + a(cos t)e2 + be3)

n = -cos t)el + ( t)e2) b = (a2 + b2)-l/2(b(sin t)el - b(cos t)ez + ae3) t, n, b e3, 3'

. t

= t ( __ b2 )/2 =

1 a2 + b2

n b Pn = 1 P b

b

4.1. = (1 + t)el - t2e2 + (1 + t3)e3 t = 1.

W '

t = 1

= x(l) + kx'(l) = (2 + k)e} - (1 + 2k)e2 + (2 + 3k)e3 .

.

( - (1' '(1) = ( - 2) + (2 + 1)(-2) + (3 - 2)3 = - 22 + 33 = 10

4.2. 2 (cos t)el + (sin t)ez + te3 (t> )

= + ' = (cos t - k t)el + ( t + k cos t)e2 + (t + k)e3 , ,

= cos t - k t, 2 = t + k cos t, 3 = t + k . 2 3 = . t + k = k = -t.

= cos t + t t, 2 = t - t cos t, 3 =

..

72

4.3.

.4

= ateI + bt2ez + t3e3, 2b2 = 3a, a = eI + e3.

" ' = ae} + 2bte2 + 3t2e3 c Ix'l = (a2 + 4b2t2 + 9t/2 = (a2 + 6at2 + 9t4)I/2 = + 3t2

2b2 = 3a. , ' c a { ' a } { + 3t2 } Cos- I -,-,,-,-, = Cos-I _n = Cos-I (1/V2) = /4 a ( + 3t2) 2

4.4. , , , ,

. = , , (_ 4.28).

xI(s*)eI + x2(s*)e2 + s*(COS a)e3 c c e3 Ec.

cos = cos 4(t, e3) = t e3 = ' e3 = 3 'Ooco 3 = COS

3 = 8 COS + c, c = .

, ~ /2, 8* = 8 + c/(COS ), 3 = 8* COS c c c

= XI(8*)eI + X2(8*)e2 + 8*(COS a)e3 . ' = /2, 3 = c = , c . = xI(s)eI + x2