CH1-4, Διαφορικη΄Γεωμετρία
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Transcript of CH1-4, Διαφορικη΄Γεωμετρία
-
1pi pipi
pi , pi, -
pi. pi pi,
pi pi pi pi pi, pi-
pi pi pi pi pi
pi pi .
1.1 pi;
pi pi, pi
, y2x D 1 ( pi!), , x2 C y2 D 1, pi, y x2 D 0:
y 2x = 1 y x2 = 0 x2 + y2 = 1
1.1.
pi pi
f .x; y/ D c;
1
-
2 1. pi pipi
pi f x y, c . pi pi
pi ,
C D f.x; y/ 2 R2 j f .x; y/ D cg: (1.1) pipi pi pi pipi R2, pi pi-
pi R3 pi, x R3
pi pi pi
y D 0; z D 0:, pi R3 pi pi pi
f1.x; y; z/ D c1; f2.x; y; z/ D c2: pi pi , pi . 1.1
pi .x; y/ pipi pi
pi f .x; y/ c.
pi pi pi pi, pi pi-
pi pi pipi: pi pi
pi . , .t/ -
t , pi pi pi
pi t ( R2 pi pipi pi,
R3 pi pi ).
pi pi Rn (
pipi n D 2 3, pi pipi).
1.1.1
pi pi Rn pi W .; / ! Rn, pi ; 1 < 1:
.; /
.; / D ft 2 R j < t < g: pi pi pi pi pi -
C pi ( ) C. pi
pi pi pi pi pi pi -
pi pi .
1.1.2
pi .t/ pi y D x2. .t/ D .1.t/; 2.t//, 1 2 pipi pi
2.t/ D 1.t/2 (1.2) t .; / pi ( pi-
pi ), pipi pi pipi
.1.t/; 2.t// pi t 2 .; /. , pi pi . 1.2: 1.t/ D t , 2.t/ D t2. pi pi pipi pi t pi pi ( .t/
-
1.1. pi; 3
pi t pi pi pi-
pi pi ), pi pipi pi .; /
.1;1/. pi pi pi :
W .1;1/! R2; .t/ D .t; t2/:
pi pi. pi
.t/ D .t3; t6/ ( .; / D .1;1/). .2t; 4t2/, pi- (pi) . pi pi pi
.
1.1.3
pipi pi x2 C y2 D 1. pi pi pi pi x D t pi pi pi, y D
p1 x2 ( pi pi pi y D
p1 x2). pi
pi pi
.t/ D .t;p1 t2/:
pi , p1 x2
pi 0. , pi y D p1 x2, pi
.
pi , pipi pipi-
. 1.t/ 2.t/
1.t/2 C 2.t/2 D 1 (1.3)
t 2 .; /, pi .1.t/; 2.t// pi t 2 .; /. pi pi . 1.3: 1.t/ D cos t , 2.t/ D sin t( cos2 t C sin2 t D 1 t). pi pi .; / D .1;1/, pi: pipi .; /
pi 2 .
pi pi pi pi pi pi
pi pi pi .
1.1.4
pi pi (pi )
.t/ D .cos3 t; sin3 t/; t 2 R: cos2 t C sin2 t D 1 t , x D cos3 t , y D sin3 t .t/ pi
x2=3 C y2=3 D 1: pi pipi pi .
. 1.1.5.
pi pi ( pi-
) pi pi . pi pi-
pi pi :
-
4 1. pi pipi
f W .; /! R pi dnfdt
pi n 1 t 2 .; /. f .t/ g.t/ , pi pi- pi pi f .t/C g.t/, f .t/g.t/, pi f .t/=g.t/ f .g.t// pi
.
pi .t/ (pi
1.1.1), pi :
.t/ D .1.t/; 2.t/; : : : ; n.t//;
d
dtDd1
dt;d2
dt; : : : ;
dn
dt
;
d 2
dt2Dd 21
dt2;d 22
dt2; : : : ;
d 2n
dt2
; ...
, d=dt P.t/, d 2=dt2 R.t/,pi. pi dn=dtn pi n 1 t 2 .; / , 1; : : : ; n .
pi pi pi
.
1.1.5
pi pi, pi pi P.t/ -pi .t/.
, pi
.t C t/ .t/t
pi pi .t/ .t C t/ C
( 1.2).
(t)
(t + t)
1.2. pi pi.
t , pi , -
pipi pi pi
C .t/.
.t C t/ .t/t
;
-
1.1. pi; 5
pi d=dt t . , pi
pipi pi pi d=dt
. , pi
C p C pi pi p pi
d=dt .
pi pi:
1.1.6
pi pi pi ,
pi ( ).
pi P.t/ D a t , pia , pi
.t/ DZ
d
dtdt D
Zadt D taC b;
pi b . a 0, pipi pi pi pi a pi b (
1.3).
(t )ta
b a
0
1.3.
a D 0, ( b). pi pi pi, pipi pi-
pi pi pi pi pi-
pi pi.
pi.
1.1.7
1 pi pi
.t/ D ..1C 2 cos t/ cos t; .1C 2 cos t/ sin t/; t 2 R( 1.4). ,
.t/ D 0 t D 2=3 t D 4=3. pi P.t/ D . sin t 2 sin 2t; cos t C 2 cos 2t/:
1... Limacon pi limax pi , .
-
6 1. pi pipi
,
P.2=3/ D .p3=2;3=2/; P.4=3/ D .
p3=2;3=2/:
pi pi pi ;
P.t/ t , pi t D 2=3 t D 4=3, pi 0 pi.
1.4. .
pi pi pi pipi pi -
pi pi pi : ,
pipi pi pi t pi,
.t/ 2 Rn. pi, 1.1.5 pipi pi : pi pi, pi
pi P.t/ pi pi t., pi, pi , pi pi
pi pi
pi pi 1.1.5.
pipi .
1.1.1 .t/ D .t2; t4/ pi pi y D x2;1.1.2 pi pi :
(i) y2 x2 D 1;(ii) x
2
4C y2
9D 1:
1.1.3 pi pi:
(i) .t/ D .cos2 t; sin2 t/;(ii) .t/ D .et ; t2/:
1.1.4 pi pi pi 1.1.3.
-
1.1. pi; 7
1.1.5 pi 1.1.4. pi
pi . pi -
pi ;
1.1.6
x2
p2C y
2
q2D 1;
pi p > q > 0 ( 1.5). Dq1 p2
q2,
.p; 0/ x , pi f1 f2. pi .t/ D .p cos t; q sin t/ pi . pi :
(i) pi p pi
f1 f2 pi p.
(ii) pi pi
p pi f1 f2 pi p.
(iii) p , pi f1 p pi f2 p
pi p.
f
pp
2f1
1.5.
1.1.7 pipi pi pi pi-
pi .
, x a > 0,
pi pi :
.t/ D a.t sin t; 1 cos t/:
1.1.8 1 .12; 0; 0/,
12 pi z.
.t/ D cos2 t 12; sin t cos t; sin t
pi pi pi -
-
8 1. pi pipi
1.6.
pi. pi pi
Viviani2 ( 1.6).
1.1.9 pi p pi
pi p pi p. pi-
pi.t/ D .2 cos tcos 2t; 2 sin tsin 2t/ pi t D =4.
1.2
pi v D .v1; : : : ; vn/ Rn,
kvk Dqv21 C C v2n:
u Rn, ku vk pi u v Rn.
pi pi pi pi-
, t pi , C .t/
.t C t/ , pi pi k.t C t/ .t/k:
pi t , ..t C t/ .t// =t pipi P.t/, pi pi
k P.t/kt: (1.4) pi ( )
C, pi , pi pi
t t , pi pi-
(1.4), pi. t
pi .
2... Vincenzo Viviani (16221703) .
Torricelli . pi 1.9 pi , Viviani
: pi , pi
pi pi .
-
1.2. 9
1.7. .
:
1.2.1
pi .t0/ s.t/
pi pi
s.t/ DZ tt0
k P.u/kdu:
pi, s.t0/ D 0 s.t/ t t0. pi .Qt0/, Qs pi s R Qt0
t0k P.u/kdu
Z tt0
k P.u/kdu DZ tQt0k P.u/kduC
Z Qt0t0
k P.u/kdu:
1.2.2
pi
.t/ D .ekt cos t; ekt sin t/;pi k ,
P.t/ D .ekt .k cos t sin t/; ekt .k sin t C cos t//;) k P.t/k2 D e2kt .k cos t sin t/2 C e2kt .k sin t C cos t/2 D .k2 C 1/e2kt :pi, .0/ D .1; 0/ ( pi-
)
s.t/ DZ t0
pk2 C 1ekudu D
pk2 C 1k
.ekt 1/:
. , s
-
10 1. pi pipi
1.8. pi.
pi .t0/,
ds
dtD d
dt
Z tt0
k P.u/kdu D k P.t/k: (1.5)
.t/ t ,
ds=dt ( pi
pi). .
1.2.3
W .; / ! Rn pi pi,
.t/ k P.t/k. pi P.t/ t 2 .; /.
pi pi pi pi
pi pi pi pi .
pi pi pi.
pi , pi .
pi a D .a1; : : : ; an/ b D .b1; : : : ; bn/ Rn
a b DnX
iD1aibi :
a b pi t , pi pi
d
dt.a b/ D da
dt bC a db
dt:
pi pipi pi pi
-
1.2. 11
d
dt.aibi / D dai
dtbi C ai dbi
dt:
1.2.4
n.t/ pi pi t ,
Pn.t/ n.t/ D 0 t , Pn.t/ n.t/ t .
, pi , R P .pi pi pi pi
n n D 1 pi t , piPn nC n Pn D 0;
2 Pn n D 0. pipi pi n D P .
1.2.1 pi .t/ D .t; cosh t/ - .0; 1/. pi
pi (. 2.2.4).3
1.2.2 pi :
(i) .t/ D13.1C t/3=2; 1
3.1 t/3=2; tp
2
;
(ii) .t/ D 45cos t; 1 sin t;3
5cos t
:
1.2.3 pipi pi pi
./ D .r cos ; r sin /;pi r ( .r; / pi
./). pi pi ;4
r./ pi . ,
, .
;
3... pi pi pi pi. pi ,
pi pipi pi, pi ,
. pi , , Jacob Bernoulli pi
pi y D a coshx, a D . coshx 1C x2
2pi pipi pi pi Taylor 2 coshx.
pi pi pi!4... . 1.3.3.
-
12 1. pi pipi
1.2.4 pi
pi pi . p q ,
pi pi pi , pi
.a/ D p, .b/ D b, pi a < b. , u ,
P u k Pk pi
.q p/ u Z ba
k Pkdt:
u D .q p/=kq pk, p q pi
kq pk.
1.3 pi
1.1.2 1.1.3 pi pi
pi pi.
.
1.3.1
pi pi Q W . Q ; Q/! Rn pi pi- pi W .; / ! Rn pi 11 pi pi W . Q ; Q/! .; / ( pi pi) pi- 1 W .; /! . Q ; Q/ pi
Q.Qt/ D ..Qt// Qt 2 . Q ; Q/: (1.6)
, pi , pi
Q:Q.1.t// D ..1.t/// D .t/ t 2 .; /:
pi pi pi ,
pi pipi .
1.3.2
1.1.3 pi x2 C y2 D 1
.t/ D .cos t; sin t/. pi
Q.t/ D .sin t; cos t/( sin2 t C cos2 t D 1). pi Q pi ,pipi pi pi
.cos.t/; sin .t// D .sin t; cos t/: .t/ D =2 t .
-
1.3. pi 13
pi pi 1.2, pi pi
. pi
pi pi pi .
1.3.3
.t/ pi pi
P.t/ 0 .t/ . pi 5 .
pi pi
, pi pi pi.
pi , pi pi
.
1.3.4
pi pi .
pi pi Q pi 1.3.1, t D .Qt/ D 1 Qt D .t/. - . .t// D t pi t pi pi
d
d Qtd
dtD 1:
d=d Qt pi . Q.Qt/ D ..Qt//, pi
d Qd Qt D
d
dt
d
d Qt ;
pi pi pipi d Q=d Qt pi d=dt pi.
1.3.5
.t/ pi, s (. 1.2.1),
, t .
pi ( ) s -
t
ds
dtD k P.t/k :
pi , pi pi-
pi pi,
.t/ D .u.t/; v.t//;5... pi regular point regular curve
. pi, singular point.
-
14 1. pi pipi
pi u v t ,
ds
dtDpPu2 C Pv2:
f .x/ D px .0;1/. , pi pi n 1
dnf
dxnD .1/n1 1 3 5 : : : .2n 1/
2nx.2nC1/=2:
u v t , Pu Pv, Pu2C Pv2. , Pu2C Pv2 > 0 t ,
ds
dtD f . Pu2 C Pv2/
t , pi s .
pi pi pi.
1.3.6
pi pi pi
.
pi pi pi pi W .; / ! Rn pi Q pi pi . t D .Qt/ Q.Qt / D .t/,
d Qd Qt D
d
dt
dt
d Qt ;
)
d Qd Qt
D
ddt
dtd Qt:
, kd Q=d Qtk D 1, pi d=dt pi .
, pi pi d=dt pi
. pi . 1.5 pi ds=dt > 0 t , pi s
pi,
1.3.5 s t . pi pi (
pi pi ) pipi s W .; /! R 11, . Q ; Q/, pis1 W . Q ; Q/! .; / . ( pi - pi pipi pipi pi pi
pi pi 1.5 pi
5.6.) D s1, Q pi
-
1.3. pi 15
Q.s/ D .t/ (. 1.6). pi
d Qds
ds
dtD d
dt;
)
d Qds
dsdt D
ddt
D dsdt ; ( . 1.5),
)
d Qds
D 1:
pi pi 1.3.6 pipi
pi pi.
1.3.7
pi Q pi :Q.u.t// D .t/ t;
pi u t . s (
),
u D s C c; (1.7)pi c . , u pi . 1.7 pi
c pipi pi, Q pi - .
pi pi pi pi pi
1.3.6 pipi u pi
du
dtD
ddt
D dsdt ( . 1.5);
pi u D s C c pi c. pi pi ,
pi pipi pi,
pi pi pi.
1.3.8
pi .t/ D .ekt cos t; ekt sin t/; 1.2.2 k Pk2 D .k2C1/e2kt : pi , . - .1; 0/ s Dpk2C1.ekt1/=k.6
t D 1kln
kspk2C1 C 1
, pi pi -
6... pi 1.2.2.
-
16 1. pi pipi
pi pi
Q.s/ D
kspk2 C 1 C 1
cos
1
kln
kspk2 C 1 C 1
;
kspk2 C 1 C 1
sin
1
kln
kspk2 C 1 C 1
:
1.3.9
7 pi
.t/ D .t; t2; t3/; t 2 R:
1.9. .
P.t/ D .1; 2t; 3t2/,
k P.t/k Dp1C 4t2 C 9t4:
pi , .
.0/ D 0
s DZ t0
p1C 4u2 C 9u4du:
pi pi pi
, . ( pi
pi .)
pi pi pi
pi.
7... pi twisted cubic.
-
1.3. pi 17
1.3.10
pi .t/ D .t; t2/ pi y D x2, P.t/ D .1; 2t/ pi pi , . Q.t/ D .t3; t6/ pi pi. PQ D .3t2; 6t3/, pi t D 0, pi Q .
1.3.1 pi pi pi ;
(i) .t/ D .cos2 t; sin2 t/ pi t 2 R.(ii) pi pi (i) 0 < t < =2.
(iii) .t/ D .t; cosh t/ pi t 2 R.
pi pi.
1.3.2 8 ( 1.10) pi pi pi -
.r; /
r D sin tan ; =2 < < =2: pi pi pi-
.t/ Dt2;
t3p1 t2
; 1 < t < 1;
pi .
1.10. .
8... (240180 pi..) pi .
pi pi ,
.
-
18 1. pi pipi
1.3.3 pi pi :
p , pi pi t0,
P.t0/ D 0 R.t0/ .t0/ - (, pipi
). :
(i) pi .t/ D .tm; tn/, pim n , .m; n/ D .2; 3/ .3; 2/.
(ii) 1.3.2
.
(iii) p, pi-
pi pi .
1.3.4 :
(i) Q pi pi , pi- Q .
(ii) Q pi O pi Q , O pi .
1.4 pi
pi pi pi , pi ,
pi , pi pi. ,
, pi pi pi , pi -
pi pi pi pi
pi . pi , pi
pi, pi pi . pi
pipi pi pi ( 1.11)
.t/ D .t2 1; t3 1/:
pi pi pi pi
pi ,
pipi pi. pi, pipi pi
pi .
1.4.1
W R ! Rn pi T 2 R. T -pi
.t C T / D .t/ t 2 R: T -pi pi T 0, .
-
1.4. pi 19
1.11. pi .
pi, T -pi, pi pi pi
pi T , pipi
. , pi 0-pi.
T -pi, pi pi pi pi
pipi jT j. , pi pi pi , W a; b ! Rn. pi pi a b,9 pi
pi pi .b a/-pi () pi W R ! Rn. pi, pi pi pi
.
pi pi T -pi .T /-pi,
.t T / D ..t T /C T D .t/:pi T -pi pi T 0, T -pi pi T > 0.
1.4.2
pi pi T pi
T -pi.
pi T pi (pi
pi ). pi
pi T pi .
9 pi a b pipi pi .
-
20 1. pi pipi
1.4.3
.t/ D .p cos t; q sin t/ ( 1.1.6) pi pi2 , 2-pi (pi pipi
pi ).10
pi, pi -
pi . pi pi
pi pi pi pi T
pi t pi pipi
T , pi 0 t T ,
`./ DZ T0
k P.t/kdt:
pi 1.3.6, pi
s DZ t0
k P.u/kdu
pi pi pi Q ( Q.s/ D .t//.
s.t C T / DZ tCT0
k P.u/kdu DZ T0
k P.u/kduCZ tCTT
k P.u/kdu D `./C s.t/;
, v D u T pi .u T / D .u/ ( pi P.u T / D P.u/), piZ tCT
T
k P.u/kdu DZ t0
k P.v/kdv D s.t/:
pi,
Q.s.t// D Q.s.t 0//() .t/ D .t 0/() t 0 t D kT () s.t 0/ s.t/ D k`./;pi k . pi pi Q pi pi `./. , Q , `./ pi Q . ,pi pi pi pi pi
.
pi pi 1.11, pi -
pi , pi
pi x < 0, pi -
. pi pi pi .
1.4.4
pi p pi pi
a b 10... T -pi pi pi .
-
1.4. pi 21
(i) .a/ D .b/ D p,
(ii) pi T , a b pipi T .
1.4.5
1.1.7 pi pi 2 . pi
, . ( pi
pi pi pi. 1.4.1 .)
1.4.1 Cayley
.t/ D .cos3 t cos 3t; cos3 t sin 3t/; t 2 R; pi . pi ;
( pi
pi 6 ).
1.4.2 pi pi pi
.
1.4.3 pi T1-pi T2-pi,
.k1T1 C k2T2/-pi pipi k1 k2.
1.4.4 W R ! Rn pi pi T0 - pi T0-pi. pi T -
pi T D kT0 pi k.
1.4.5 pi W R ! R T -pi pi T 0. pi pi T0 pi T0-pi.
pi pi pi pi pi .
pi pi pi, pi pi -
T0.
(i) pi T1; T2; T3; : : : T1 > T2 >
T3 > > 0 Tr -pi r 1.(ii) fTrg (i) pi pi
Tr ! 0 r !1.(iii) pi pi fTrg pi (i)
Tr ! 0 r !1 pi .
1.4.6 W R ! Rn pi pi T -pi piT > 0. .
-
22 1. pi pipi
1.5 pi pi
pi
pipi pi
pi pi pi .
pi , pi pi
pi pi. pi,
pi x2 C y2 D 0 . pi pipi pi f .x; y/ f .x; y/ D c, pi c , pi pi pipi pi
, pi pi pi pi
pi. pi pi c D 0 (pi f f c).
1.5.1
f .x; y/ (pi
pi f , , pi ). pi
, pi
C D f.x; y/ 2 R2 j f .x; y/ D 0g; @f=@x @f=@y . p C, -
.x0; y0/, pi pi pi .t/, -
pi pi 0, pi pi p t D 0 pi C t .
pi pi
pi ( pi pi pi
1.3.6). pi, pipi pi pi
. pi ( 5.6.2), -
pi pi pi-
pi.
pi pipi pi f
1.5.1, pi .x0 Cx; y0 Cy/ C p
f .x0 Cx; y0 Cy/ D 0: Taylor ,
f .x0 Cx; y0 Cy/ D f .x0; y0/Cx@f@x
Cy @f@y
;
pi pi pix y ( pi
pi .x0; y0/). pi,
x@f
@xCy @f
@yD 0: (1.8)
x y , .x;y/ pi C
-
1.5. pi pi pi 23
p, . 1.8 n D@f@x; @f@y
C
p ( 1.12).
(x, y)
p
n
C
x
y
1.12.
pi 1.5.1 n
C. pi pi @f@y ; p, n
pi x p, pi pi C p
pi y ( 1.13).
x
p
x0
y0
y
C
1.13.
x D x D x0 C .x; y/ p. ,
f .x; y/ D 0 (1.9) y y0 x x0.
-
24 1. pi pipi
pi pipi pi pi C p pi-
y ( @f=@y D 0), pi .. 1.14.y
p
xx0
C
1.14. pi pi y 0 p.
pi x D x D x0 C p, x D x0 C pi p.
f pi pi pi pi pi-
pi g.x/, x pi -
x0, y D g.x/ . 1.9 x0. pi pi pi C p
:
.t/ D .t; g.t//: g ( pi pipi pi
pi), P D .1; Pg/ pi pi. pi pi 1.5.1.
x2 + y2 = 1 x2 y2 = 1
1.15. pi.
pi pi pi pi pi -
pi 1.5.1. pi f .x; y/ pi
-
1.5. pi pi pi 25
pi pipi pi C pi pi
f .x; y/ D 0 . pi pi, C pi pi .
pi, x2Cy2 D 1 , pi x2y2 D 1 (. 1.15). pi f , pi pi-
pi pi C. pipi, C
, pi pi 11 C , pi-
pi ; pi C, ./ D ./ 11 .; /.
pi pi pi pi-
pi :
1.5.2
pi pi pipi .t0/ D .x0; y0/ , pi , pi f .x; y/,
x y pi pi x0 y0,
, pi pi 1.5.1 .t/
pi pi f .x; y/ D 0 t pi pi pi t0.
pi 1.5.2 pi pi
1.5.1.
.t/ D .u.t/; v.t//;pi u v . , pi
Pu.t0/ Pv.t0/ , Pu.t0/. u t pi t t0 ( 1.16).
u
u0
t0t
1.16.
pi pi 1.5.1, pi
t u D x0 x u.t/,
-
26 1. pi pipi
pi t t0. pi pipi h.x/, -
x pi pi x0, t D h.x/ u.t/ D x x x0 t t0. pi h .
f .x; y/ D y v.h.x// pi pi.
f .x; y/ pi -
pi 1.5.1 pi-
pi f .x; y/ D 0, pi pi 1.1.7. pi pi
pi pi f pi pi -
1.5.1 pi pi
.
1.5.1 pi C
y2 D x.1 x2/ . pi t pi C
.t/ D .t;pt t3/I
pi;
1.5.2 pi 1.5.1 pi R3
pi pi f .x; y; z/ D g.x; y; z/ D 0.1.5.3 pi pi 1.5.2 pi
R3 ( Rn). ( .)
pipi pi pi,
pi pi
pi pi ( pi) .
-
2 pi pi;
pi R3 -
, pi . pi pi pi
pi pi ( pi-
), pi pi pi
pipi ( pipi pi ). pi pi-
, pi pi pi.
2.1 pi
pipi pi pi -
pi. pipi pi pipi pi.
pipi pi, pi
pipi pi p pipi pi pi
pi p.
pi pi pi R2.
pi t t Ct , pi pi pi -pi .t/ pi ..t Ct/.t// n, pi n pi P.t/ .t/.
Taylor,
.t Ct/ D .t/C P.t/t C 12R.t/.t/2 C pipi; (2.1)
pi pi .pipi/=.t/2 t .
P n D 0, pi pi pi .t/
27
-
28 2. pi pi;
(t)
(t + t)
n
2.1. pi pipi pi pi pi .
1
2R.t/ n.t/2 C pipi:
, R pi- n. pi, pi pipi, pi-
pi pi
1
2k R.t/k.t/2:
pi :
2.1.1
pi pi t , pi .t/
.t/ pi k R.t/k. pi pi
Rn n 2. pipi, pi 1.1.6,
pi R D 0 pi, pipi pi.
pi pi pi pi
2.1.1 pi pi . -
R2 .x0; y0/ R, pi
:
.t/ Dx0 CR cos t
R; y0 CR sin t
R
:
-
2.1. pi 29
P.t/ D sin tR; cos t
R
,
k P.t/k Ds
sin tR
2Ccos
t
R
2D 1;
pi pi . pipi,
R.t/D 1R
cost
R; 1
Rsin
t
R
;
pi pi
k R.t/k Ds
1R
cost
R
2C 1R
sint
R
2D 1
R;
. pi
pi , pipi pi
pi.
pi .
pipi pi, , 1.3.6,
pi , Q , pi pi -pi pi Q . , pipi O pi , pi Q O . pi pi O pi- Q ( 1.3.4), pi, 1.3.7,
Q.t/ D O.u/;pi u D t C c c . , , pi
d Qdt
D d Odu
,
d 2 Qdt2
D ddu
ddu
D d
2 Odu2
;
pi Q O pi pi. pi pi ,
pi pipi pi (.
1.3.8 1.3.9). pi pi pi
pi pi .
2.1.2
.t/ pi R3, pi
D k R Pkk Pk3 ; (2.2)
pi ( ) d=dt .
-
30 2. pi pi;
, pi R2 pi pi pipi
() xy R3, . 2.2 pi pi pi pi
pi pipi pi.
pi s pi .
pi
P D ddt
D dds
ds
dt;
pi
D
d 2ds2
D
ddsd=dt
ds=dt
D
ddt
d=dtds=dt
ds=dt
D
dsdt
d2
dt2 d2s
dt2ddt
.ds=dt/3
: (2.3)pipi,
ds
dt
2D k Pk2 D P P ;
pi pi t pi
ds
dt
d 2s
dt2D P R :
pi . 2.3 pi
D
dsdt
2R d2s
dt2dsdtP
.ds=dt/4
Dk. P P/ R . P R/ Pk
k Pk4 :
pi
a .b c/ D .a c/b .a b/c(pi a;b; c 2 R3) pi
P . R P/ D . P P/ R . P R/ P:pipi, P R P , pi
k P . R P/k D k Pkk R Pk:pi,
k. P P/ R . P R/ Pkk Pk4 D
k P . R P/kk Pk4 D
k Pkk R Pkk Pk4 D
k R Pkk Pk3 :
pi . 2.2 P 0. pi pi pi.
-
2.1. pi 31
2.1.3
z pi
./ D .a cos ; a sin ; b/; 2 R;pi a b .
2.2. .
.x; y; z/ ,
x D a cos ; y D a sin ; z D b pi , x2 C y2 D a2, pi pi pi jaj z. jaj . 2 , .a cos ; a sin ; b/ pi-
pi z pi
2b. 2jbj . pi pi pi pi -
2.1.2. d=d , P./ D .a sin ; a cos ; b/,
k P./k Dpa2 C b2:
pi P./ pi , ( a D b D 0 pipi ). pi pi
pi 2.1.2: R D .a cos ; a sin ; 0/, R P D .ab sin ; ab cos ;a2/ pi
D k.ab sin ; ab cos ;a2/k
k.a sin ; a cos ; b/k3 D.a2b2 C a4/1=2.a2 C b2/3=2 D
jaja2 C b2 : (2.4)
pi .
-
32 2. pi pi;
pipi pi
pi . , pi b D 0 ( a 0). pipi pi pipi xy jaj,pi pi pi 2.1.2 pi
1=jaj. pi , . 2.4 pi
D jaja2 C 0 D
jaja2
D 1jaj :
pi pi a D 0 ( b 0). pipi z, , pi pi pi
. 2.4 pi pi a D 0.
2.1.1 pi pi pi pi:
(i) .t/ D13.1C t/3=2; 1
3.1 t/3=2; tp
2
.
(ii) .t/ D 45cos t; 1 sin t;3
5cos t
.
(iii) .t/ D .t; cosh t/.(iv) .t/ D .cos3 t; sin3 t/. (iv), pi 1 pi pi .1; 0/, .0;1/. pi pi 1.1.5.
2.1.2 , pi .t/ pi .t/ pi
> 0, .t/ t . pi
pi pi > 0.
2.2 pipi pi
pi pi pipi pipi
pi pi .
pi .s/ pi R2. -
d=ds ,
t D P pi t .pi
t pi pi-
ns pi pipi
pi t =2 ( 2.3).
1.2.4, Pt D R t pi ns .pi pi s
R D sns :
-
2.2. pipi pi 33
ns t
2.3. pi pi pipi pi.
s pi pi (pi ,
). , knsk D 1, D k Rk D ksnsk D js j; (2.5)
pi pi pi pi-
. 2.4 pi pi pi pi pi
pi ( pipi pi pi pi
s). s
.
s > 0
ns
ns
ns
ns
< 0 < 0 > 0
t
t
t
t
s s s
2.4. pi s .
.t/ , pi,
pi t, pi -
ns pi pi s
pi Q.s/, pi s . pi,
t D d=dtds=dt
D d=dtkd=dtk ;
-
34 2. pi pi;
ns pi t =2 ,
d t
dtD d t
ds
ds
dtD s
ds
dtns D s
ddt
ns :
pi pi pi -
pi pi . pi
, pi P.s/ pi '.s/ pi
P.s/ D .cos'.s/; sin '.s//: (2.6), '.s/ , pi pi pi
pi pi pipi 2 . pi
pi pi pi ':
2.2.1
W .; / ! R2 pi , s0 2 .; /, '0
P.s0/ D .cos'0; sin '0/: pipi pi ' W .; /! R '.s0/ D '0, pi . 2.6 s 2 .; /.
pi
P.s/ D .f .s/; g.s//:
f .s/2 C g.s/2 D 1 s; (2.7) .
'.s/ D '0 CZ ss0
.f Pg g Pf /dt:
'.s0/ D '0. pipi, f g , P' D f Pg g Pf '.
F D f cos' C g sin ' G D f sin' g cos':pi
PF D . Pf C g P'/ cos' C . Pg f P'/ sin ';pi
Pf C g P' D Pf .1 g2/C fg Pg D f .f Pf C g Pg/ D 0;pi pi . 2.7
pi
f Pf C g Pg D 0: pi pi Pg f P' D 0. pi PF D 0 F . pi G .
F.s0/ D f .s0/ cos'0 C g.s0/ sin '0 D cos2 '0 C sin2 '0 D 1;
-
2.2. pipi pi 35
pi pipi G.s0/ D 0. pi f cos' C g sin' D 1; f sin ' g cos' D 0
s. pi pi f D cos', g D sin ', ' pi . 2.6.
,
.s0/ D '0 P.s/ D .cos .s/; sin .s// s 2 .; /, pi n.s/
.s/ '.s/ D 2n.s/ s 2 .; /:pi ' , n s.
n . , pi n.s0/ n.s1/ pi s1 2 .; /, pi, , n.s/ pipi pi n.s0/ n.s1/
s s0 s1. pi pi
n.s0/ n.s1/ ! pi, pi n
pi s, .s0/ D '.s0/ D '0, pipi n D 0 .s/ D '.s/ s 2 .; /.
2.2.2
' 2.2.1 pi
pi '.s0/ D '0. pi pi
pi pi pi .
2.2.3
.s/ pipi pi '.s/ pi
,
s D d'ds
:
pi, pi pi pi pi
pi. pi 2.4, pi pi
t pi
pi pi
s.
pi . 2.6 pi t D .cos'; sin '/,
Pt D P'. sin '; cos'/:pi ns D . sin '; cos'/, pi pipi pi Pt D sns .
-
36 2. pi pi;
2.2.4
pi pi pi ( 1.2.1).
pi pi .t/ D .t; cosh t/ pi P D .1; sinh t/
s DZ t0
p1C sinh2 tdt D sinh t:
pi, ' pi P x, tan' D sinh t D s;
) sec2'd'
dsD 1;
) s D d'ds
D 1sec2'
D 11C tan2 ' D
1
1C s2 :
2.2.3 pi pi pi
pi pi `, -
Z `0
s.s/ds: (2.8)
2.2.5
pi pi pipi pi
pipi 2 .
pi pipi pi `
. 2.2.3 pi pi
Z `0
d'
dsds D '.`/ '.0/;
pi ' pi . `-pi (. 1.4):
.s C `/ D .s/: pi
P.s C `/ D P.s/; P.`/ D P.0/. pi, . 2.6,
.cos'.`/; sin'.`// D .cos'.0/; sin'.0//;pi pi '.`/ '.0/ pipi 2 .
pipi , pi pi
pipi pi ,
-
2.2. pipi pi 37
pi pi.
R2 R2 piM W R2 ! R2 M D Ta ;
pi pi pi
,
.x; y/ D .x cos y sin ; x sin C y cos /; Ta a,
Ta.v/ D vC a; pipi .x; y/ v R2 (. 1).
2.2.6
k W .; /! R , pi pi W .; /! R2 pi pi pi k.
pipi, Q W .; / ! R2 pi pi pi pi k, pi M
R2
Q.s/ D M..s//; s 2 .; /:
pi pi pi s0 2 .; / s 2 .; /
'.s/ DZ ss0
k.u/du; (pi. 2.2.3),
.s/ DZ s
s0
cos'.t/dt;
Z ss0
sin '.t/dt
:
pi pi
P.s/ D .cos'.s/; sin '.s//; pi pi ' x. pi,
, 2.2.3, pi
pi
d'
dsD d
ds
Z ss0
k.u/du D k.s/:
, Q'.s/ pi Q . pi,PQ.s/ D .cos Q'.s/; sin Q'.s//;
) Q.s/ DZ s
s0
cos Q'.t/dt;Z ss0
sin Q'.t/dtC Q.s0/: (2.9)
2.2.3 k.s/ D d Q'=ds,
Q'.s/ DZ ss0
k.u/duC Q'.s0/:
-
38 2. pi pi;
pipi . 2.9 a
Q.s0/ Q'.s0/, pi
Q.s/ D TaZ s
s0
cos.'.t/C /dt;Z ss0
sin.'.t/C /dt
D Tacos
Z ss0
cos'.t/dt sin Z ss0
sin'.t/dt;
sin
Z ss0
cos'.t/dt C cos Z ss0
sin '.t/dt
D TaZ s
s0
cos'.t/dt;
Z ss0
sin '.t/dt
D Ta ..s//:
2.2.7
pipi pi pi pi
. pi pi pi
pi s pi pi , pi .
2.5
s D :A priori, pi s D pi pi s D pi , pi s
s (. 2.2.2). , spi pipi pi .pi, s D pi s D pi. , s .
pi pipi , pi s, pi-
pi pi pi pi
s . , pi pi pi-
pi s pi pi
. pi pi
pi , pi pi pi pi
pi ( ).
pi
R
.s/ DR cos
s
R;R sin
s
R
:
pi
t D P.s/ D sin s
R; cos
s
R
;
pi =2 C s=R x( 2.5).
-
2.2. pipi pi 39
t
s = R
xs = R
2.5. pi R.
pi pi
d
ds
2C s
R
D 1
R:
pi, s > 0, 1=s pi pi s .
s < 0, pi pi
Q.s/ DR cos
s
R;R sin s
R
( pi pi pi -
R) pi pi 1=R. pi, R D 1=spi pi pi pi s.
pi pipi pi pi
pi ( pi pi) pi
2.1.1.
2.2.8
2.2.6, pi pipi pi pi
pi pi pipi .
pi pi pi pipi pi. pi-
, pi pi s.s/ D s. pi 2.2.6 pi s0 D 0, '.s/ D
R s0udu D s2
2,
.s/ DZ s
0
cos
t2
2
dt;
Z s0
sin
t2
2
dt
:
pi -
. (
, pi Fresnel,1 pi pi
1... Augustin-Jean Fresnel (17881827)
-
40 2. pi pi;
2.6. pi Cornu.
Cornu,2 pi pi Euler). 2.6
pi .
2.2.6 -
pi pi pi. pi
( ) pi k 0, pipipi pi pi pi k
pi k. 2.2.6 pi
. pi pi () pi pi pipi
y
x
y
x
2.7. 2.2.6 pi.
pi. Fresnel pi , pi,
pi Fresnel pi pi .2... Marie Alfred Cornu (18411902) .
-
2.2. pipi pi 41
x ( ) 1 x 1 pipi - pipi. ( pi pi pi pi
pi pi 9.4.3). ,
pi pi x 0 pi x. pi pi , pi pi -
. . 2.2.3 2.2.6 pi
pi pi pi.
2.2.1 , pipi pi ,
Pns D st:
2.2.2 pi pi pipi pi
.t/ t . ( -
2.1.2.)
2.2.3 Q pipi pi. , Q pipi pi
M R2, pi pi-
s Qs Q M , Qs D s M (, Q pi). , , Q pi- pi, Q pipi pi R2.
2.2.4 k pi pi pipi pi C
. , Ca C
v 7! av (pi a ), pi- pi Ca s 1aksa
.
pi pi pi
pipi pi C. , s C
pi , ' pi pi C
, T pi,
T cos' D ; T sin' D s;pi ; (pi
). pipi C
s D 1a
1C s
2
a2
1;
pi a D =, pi C pi pi pi 2.2.4
pipi.
-
42 2. pi pi;
2.2.5 .t/ pipi pi . pi
pi pi
.t/ D .t/C ns.t/: , s.t/ 1 t , -pi pi pi s=j1 s j.
2.2.6 pi pi pipi pi-
.s0/ -
pi . pi pi
pi .
, pi
pi .s0/ .s0 s/ pi
.s0/ D .s0/C 1s.s0/
ns.s0/
t . C .s0/ pi pi
.s0/ 3 .s0/, .s0/
pi .s0/. C, pi
pi .s0/, 1=js.s0/j D 1=.s0/, pi pi .
2.2.7 pi pi ,
pi pi, pi
( pipi pi,
pi ). pi
Ps.s/ 0 s ( d=ds), Ps > 0 s ( pi, pi pi
s pi s). 1s.s/
( pi-
pi ) pi pi pi
. pi pi
( pi
).
.t/ D a.1 sin t; 1 cos t/; 0 < t < 2;pi a > 0 ,
.t/ D a.t C sin t;1C cos t/(. 1.1.7) , pi, pi pi-
pi pipi.
2.2.8 ` pi .0/ pipi pi -
.s/. pi pi
pi pi , pi
.s/ D .s/C .` s/ P.s/;3... pi osculating circle.
-
2.3. pi 43
pi 0 < s < ` d=ds. pi
( pipi pi, -
pi
). pi pi pi s pi
, s.s/ > 0 s. pi pi
1=.` s/.
2.2.9
.t/ D .t; cosh t/ l D 0 (. pi ) 4
x D cosh11
y
p1 y2:
. 8.3 pi pi .
2.2.10 pipi pi .s/
pi ` pi pi a,
pi ` p D .0/. q pi . .s/ pi q pi-
pi pi ` s (
). .s/ pi
a pi P . .s/ D p C saC .s/.q .s//;
pi ' pi pi '.
pipi P .s/ .s/.q .s// D 0:
pi
pi pi pi pi `. .
1.1.7 pipi.
2.3 pi
pi ( pi) R3, -
pi . pipi pi pi
pi (. 2.2.6), pi .
pi, 1 pipixy a D b D 1=2(. 2.1.3) pi 1 pi, pi
pi pi pi pi
pi R3. pi pi
, pi , pi pi
pi pi R3.
4... tractrix pi trahere pi , .
-
44 2. pi pi;
.s/ pi R3 t D P pi . pi .s/ ,
.s/
n.s/ D 1.s/
Pt.s/: (2.10)
kPtk D , n . pipi, 1.2.4 t Pt D 0, t n .pi
b D t n (2.11) , t n. b.s/
.s/. pi ft;n;bg 5 R3, pi ,
b D t n; n D b t; t D n b:
n
b
t
2.8. Frenet.
b.s/ s, Pb b. pipi pi pi -
u v pi s:
d
ds.u v/ D du
ds vC u dv
ds:
( pi pi pi -
pi pi -
). pi b D t n piPb D Pt nC t Pn D t Pn; (2.12)
pi (2.10) n Ptn D nn D 0:pi . 2.12 pipi Pb t. t b, Pb pipi pi n,
Pb D n; (2.13)5... Frenet.
-
2.3. pi 45
pi , pi ( pi-
pi pi
). pi
.
, pi
pi . pi pipi pi-
, pi , pipi pi pi-
pi pi
u D s C c;pi c . pi pi
pipi pi pipi:
t 7! t; Pt 7! Pt; n 7! n; b 7! b; Pb 7! Pb:pi . 2.13 pi 7! . pi, pi pi.
pi pi 2.1.2,
pi pi
pi :
2.3.1
.t/ pi R3 pi pi,
pi pi
D . P R/ k P Rk2 ; (2.14)
pi d=dt .
, pi, .t/
.t/ pi pi pi .t/ ,
2.1.2 pi pipi
pi 2.14.
pi pi pi . 2.14 pi pi
pi 2.1.2. pi
pi pi, pi pi . 2.14
pipi pi.
pipi pi . -
pi . 2.11 2.13 pi
D n Pb D n .t n/PD n .Pt nC t Pn/ D n .t Pn/: n D 1
Pt D 1
R , pi
D 1R P d
dt
1
R
D 1R P
1
P
2R
D 12 . P R/;
R . P R/ D 0 R . P / D . P R/. pipi . 2.14, , , P R , pi
k P Rk D k Pkk Rk D k Rk D :
-
46 2. pi pi;
pipi, s pi ,
d
dtD ds
dt
d
ds;d 2
dt2Dds
dt
2d 2
ds2C d
2s
dt2d
ds;
d 3
dt3Dds
dt
3d 3
ds3C 3ds
dt
d 2s
dt3d 2
ds2C d
3s
dt3d
ds:
pi,
P R Dds
dt
3 d
ds d
2
ds2
;
. P R/ Dds
dt
6 d 3
ds3d
ds d
2
ds2
:
D
d3
ds3dds d2
ds2
dds d2ds2
2 D
. P R/k P Rk2 :
2.3.2
pi ./ D .a cos ; a sin ; b/ pi - 2.1.3.
P./ D .a sin ; a cos ; b/; R./ D .a cos ;a sin ; 0/;./ D .a sin ;a cos ; 0/; P R D .ab sin ;ab cos ; a2/;
k P Rk2 D a2.a2 C b2/; . P R/ D a2b;
D . P R/ k P Rk2 Da2b
a2.a2 C b2/ Db
a2 C b2 :
2.3.2 -
b D 0 pipi pi pipixy. , pi
pi pi pi.
2.3.3
pi R3 pi pi (
), pi pipi
pi.
-
2.3. pi 47
pi pi pi ( pi
pi pi , pi
pi pi pipi). pi-
s d=ds , .
pi pi pipi v N D d , piN , d v 2 R3. pi pi N . N D d pis pi
t N D 0; (2.15)) Pt N D 0 ( PN D 0);) n N D 0 ( Pt D n);) n N D 0 ( 0): (2.16)
pi 2.15 2.16 pipi t n N . pi
b D t n pi N . N b b.s/ ( ) s, pipi
b.s/ D N s b.s/ D N s. pipi b . pi Pb D 0, D 0.
, pi D 0 pi. pi . 2.13 pipi Pb D 0, b . pi pi pi pi- pipi pi pipi v b D . pi
d
ds. b/ D P b D t b D 0;
pi pi pipi b () , d . pi pi pipi v b D d .
pi pi pi . -
, pi
Pt D n Pb D n(pi pi n ),
pi Pn. . ft;n;bg - R3,
t n D b; n b D t; b t D n:
Pn D Pb t C b Pt D n t C b n D t C b: pipi :
2.3.4
pi R3 pi pi-
-
48 2. pi pi;
,
Pt D nPn D t C b (2.17)Pb D n:
2.17 Frenet-Serret. pi0@ 0 0 0 0 0
1A pi Pt, Pn Pb t, n b -, .
pi . ( pi -
2.3.6.)
pi Frenet-Serret:
2.3.5
pi R3 pi
, pi ( ).
pi pi pi pi -
2.2.7 2.3.3, pi pi pi -
pi pi,
pipi pi .
pi 2.3.3, b -
pi pipi, , b.
d
ds
C 1
n
D t C 1
Pn D 0;
pi pi pi
Frenet-Serret
Pn D t C b D t; ( D 0)( pi C 1
n 2.2.6). pi,
C 1n , a,
k ak D
1 n
D 1 : pi , S, a
1=. S , C, pi
pi C. r C, D 1=r , pi r D 1= pi S. pi, C S, pi a S. a C
v b D a b.
-
2.3. pi 49
2.2.6 -
pi .
2.3.6
.s/ Q.s/ pi R3 pi- .s/ > 0 .s/ s, pi M
R3
Q.s/ DM..s// s:pipi, k t k > 0 pi, pi pi
R3 pi pi k t .
pi t;n b pi ,
, Qt; Qn Qb Q . s0 pi s, t.s0/ Qt.s0/, pi pi pi pi
, .t.s0// D Qt.s0/. On D .n.s0// Ob D .b.s0//. pi ' On Qn.s0/. 0 pi ' pi pi pi
pi Qt.s0/, 0 Qt.s0/ pi On Qn.s0/. pi ft.s0/;n.s0/;b.s0/g fQt.s0/; Qn.s0/; Qb.s0/g - R3, 0 pi t.s0/;n.s0/ b.s0/ Qt.s0/; Qn.s0/ Qb.s0/, . , M M D T Q.s0/.s0/ 0 . 2.3.5, pi DM./ , T , N B pi
, ,
.s0/ D Q.s0/; T .s0/ D Qt.s0/; N .s0/ D Qn.s0/; B.s0/ D Qb.s0/: (2.18) pipi pi:
A.s/ D Qt T C Qn N C Qb B: . 2.18 A.s0/ D 3. pi , Qt T , Qt T 1, Qt D T . Qn N Qb B. pi A.s/ 3, Qt D T , Qn D N Qb D B. , pi pi A , pi Qt D T , PQ D P , pi Q.s/ .s/ . pi 2.18, pipi , Q D .
pi, pi pi A
. pi Frenet-Serret pi
PA D PQt T C PQn N C PQb B C Qt PT C Qn PN C Qb PBD Qn T C . Qt C Qb/ N C . Qn/ B C Qt NCQn .T C B/C b .N /;
pi .
-
50 2. pi pi;
, pi pi
pi
PT D kN ; (2.19)PN D kT C tB; (2.20)PB D tN (2.21)
T .s/;N .s/;B.s/ T .s0/;N .s0/;B.s0/
i D .1; 0; 0/, j D .0; 1; 0/, k D .0; 0; 1/, . pi 0@ 0 k 0k 0 t
0 t 0
1A pi PT ; PN PB T ; N B -, T ; N B s
(. 2.3.6).
.s/ DZ ss0
T .u/du:
P D T , , T , -. pi, PT D kN . 2.19, , N , k pi N . ,
B T N , B D T N , pi s pi 1 s. k D i j , .s0/ D 1, .s/ D 1 s. B
, t , . 2.21, .
2.3.1 pi ; ; t; n b pi pi pi
pi pi Frenet-Serret:
(i) .t/ D13.1C t/3=2; 1
3.1 t/3=2; tp
2
.
(ii) .t/ D 45cos t; 1 sin t;3
5cos t
.
pi (ii) ,
pipi pi .
2.3.2 pi R3 pi pi > 0
.
2.3.3 pi R3 pi > 0
pi -
a. pi
pi D cot . ,
-
2.3. pi 51
pi pi pi D , pi , pi .
pipipi, 2.1.3 2.3.2 -
. pi pi
.
2.3.4 .t/ pi .t/ > 0 .t/ 0 t . , , pi
pi ,
D d
ds
P2
: (2.22)
, . 2.22
2 C . P/2 D r2
pi () r , pi D 1= D 1= , pi pi r . pi . 2.22
pi Viviani ( 1.1.8).
2.3.5 P 33 pi a 2 R3, M.v/ D PvCa R3 (. 1). -
pi R3, pi DM./ pi - . pi , t, n, b T , N , B pi-
, ,
T D P t, N D Pn B D Pb.2.3.6 .aij / 3 3 pi ( aij D aj i
i; j ). v1; v2 v3 pi s pi pi
Pvi D3X
jD1aijvj ;
i D 1; 2 3. pi pi pi s0 - v1.s0/; v2.s0/ v3.s0/ .
v1.s/; v2.s/ v3.s/ s.
pipi
pi pi .
-
3 pi
pi pi pi,
pi pi pi
, pi pi.1 pi -
pi, pi pi pi pi
pi pi , pi
pi. pi -
pi, pi pi-
pi pi pi
.
3.1 pi pi
pipi pi :
3.1.1
pi pi R2 pi R2 .
pi, pi pi R2, pi
pi Jordan, pi pi
pipi : , pi
( R2 pi )
pi R2, pi int./ ext./,
:
1... pi pi pi
global.
53
-
54 3. pi
(i) int./ , pi -
.
(ii) ext./ .
(iii) int./ ext./ ,2 -
: pipi pi pi
pi pi ( pi pi
int./ ext./ pipi pi ).
3.1.2
.t/ D .p cos t; q sin t/, pi p q , pi pi pi 2 . -
, , n.x; y/ 2 R2 j x2
p2C y2
q2< 1
o
n.x; y/ 2 R2 j x2
p2C y2
q2> 1
o.
pi
pi pi pi pi. .. p
pi pi pi pi
3.1;
p
3.1. p pi;
3.1.3
1.1.7 pi pi,
. 3.1.1.
pi pi pi-
pi pi .
pi pi ns
2... pi .
: pi R2 ( Rn )
pi ( ) pi . pi
. .
pi pi pi.
-
3.1. pi pi 55
(. 2.2) pi int./ . pi pi
pi pi t t pi. 3.2 pi pi.
pi pi pi 3.1;
t
ns
ns
t
3.2. pi pi pi.
pi pi pi-
.
3.1.4 ( pi Hopf3)
pi pi pi pi R2 2 . pi 3.1.4 pi pi
pi pi . pi
( pi) 13.1.
2.2.5 pi pi-
pi R2 pipi 2 .
Hopf pi pi ,
pipi 1. pi pi 2.2.5 pi - Hopf ' pi pi
2 pi - pi pi 2 . pi pi pi 3.1,
pi .
3.1.1
.t/ D ..1C a cos t/ cos t; .1C a cos t/ sin t/ ;pi a , pi pi jaj < 1, jaj > 1 pi pi R2, pi pi .
a D 1;3... Umlaufsatz pi
pi. Heinz Hopf (18941971) pi pi.
-
56 3. pi
3.2 pi
pi pi pi pi
A./ D
int./
dxdy: (3.1)
pi pi pi .
Green. f .x; y/ g.x; y/ (
pi ), pi
pi pi, int./
@g
@x @f@y
dxdy D
Z
f .x; y/dx C g.x; y/dy:
pi Green pi pi -
pi .
3.2.1
.t/ D .x.t/; y.t// pi pi pi R2 pi T ,
A./ D 12
Z T0
.x Py y Px/dt: (3.2)
pi f D 12y g D 1
2x Green,
A./ D 12
Z
xdy ydx;
pi pi pipi . 3.2.
, pi pi . 3.2 pi pi t , pi
3.1.1 A./ pi pi-
.
pi pi pipi pi pi-
.
3.2.2 (pi )
pi pi, `./ A./ pi
pi pi,
A./ 14
`./2;
.
pi , pi-
pi `./ D 2R A./ D R2, pi R . pi pi pi :
-
3.2. pi 57
3.2.3 ( Wirtinger)
F W 0; ! R F.0/ D F./ D 0, Z 0
dF
dt
2dt
Z 0
F.t/2dt;
F.t/ D D sin t t 2 0; , piD .
pi pi Wirtinger, pi pi
pi pipi pi .
pi pi pi pi pipi-
pi. pi, pi, pi
pi pi s. , pi
3.2.2, pi pi . -
pi pi s
t D s`./
; (3.3)
pipi pi pi pi pi , s
`./, t . pi, pi
pi pi t . 3.3.
pipi, pi `./ A./ pi -
.t/ 7! .t/ C b, pi b (. 3.2.1). b D .0/, pi pi- pi .0/ D 0, .
pi 3.2.2 pi `./ A./ -
pi pi :
x D r cos ; y D r sin :pi pi
Px2 C Py2 D Pr2 C r2 P2; x Py y Px D r2 P;pi d=dt . pi, pi . 3.3 pipi
Pr2 C r2 P2 Ddx
dt
2Cdy
dt
2D
dx
ds
2Cdy
ds
2!ds
dt
2D `./
2
2; (3.4)
pi .dx=ds/2 C .dy=ds/2 D 1. pipi, pi . 3.2
A./ D 12
Z 0
.x Py y Px/dt D 12
Z 0
r2 Pdt: (3.5)
pi 3.2.2 pipi
`./2
4A./ 0;
-
58 3. pi
. pi . 3.4 pipi Z 0
.Pr2 C r2 P2/dt D `./2
:
, pi . 3.5, pi
`./2
4A./ D 1
4
Z 0
.Pr2 C r2 P2/dt 12
Z 0
r2 Pdt D 14I;
pi
I DZ 0
.Pr2 C r2 P2 2r2 P/dt: (3.6)
pi, pi 3.2.2, pipi I 0, I D 0 .
pi pi pipi
I DZ 0
r2. P 1/2dt CZ 0
.Pr2 r2/dt: (3.7)
pi . 3.7 pi 0, 0 Wirtinger (pi F D r : pi- r.0/ D r./ D 0 pi .0/ D ./ D 0). pi I 0. pi, . 3.7 0, I . pi -
P D 1 t , r D D sin t pi D (pi Wirtinger). D t C, pi , pi r D D sin. /. pi pi pi D, pi
pi 3.2.2 (. 3.3).
r
3.3. D: r D D sin. /.
pi pi Wirtinger.
-
3.2. pi 59
G.t/ D F.t/= sin t . d=dt , Z 0
PF 2dt DZ 0
. PG sin t CG cos t/2dt
DZ 0
PG2 sin2 tdt C 2Z 0
G PG sin t cos tdt CZ 0
G2 cos2 tdt:
4
2
Z 0
G PG sin t cos tdt D G2 sin t cos t j0 Z 0
G2.cos2 t sin2 t/dt
DZ 0
G2.sin2 t cos2 t/dt;
Z 0
PF 2dt DZ 0
PG2 sin2 t CZ 0
G2.sin2 t cos2 t/dt CZ 0
G2 cos2 tdt
DZ 0
.G2 C PG2/ sin2 tdt DZ 0
F 2dt CZ 0
PG2 sin tdt;
pi Z 0
PF 2dt Z 0
F 2dt DZ 0
PG2 sin tdt:
pi 0 PG D 0 t , G.t/ , D, t , pi pi F.t/ D D sin t .
3.2.1 `./ A./ pi
.
3.2.2 pi
x2
p2C y
2
q2D 1
(pi p q ), pi Z 0
qp2 sin2 t C q2 cos2 tdt 2ppq;
p D q.4 pi G pi ( pi
G.t/2 sin t cos t pi ). G.t/ t D 0 , F .t/= sin t 0=0 . pipi G pi pi pi. pi lHospital.
-
60 3. pi
3.3
pi pi pi -
pipi pi.5 pi pi
int./ , pi
pipi int./ pi int./.
3.4. pi.
3.3.1
pi .t/ R2 pi -
pi s, pi ds=dt D 0.pi pi pi,
pi .
3.3.2
pi pi pi, pi pi
.t/ D .p cos t; q sin t/, pi p q ,
s.t/ D pq.p2 sin2 t C q2 cos2 t/3=2 :
pi
ds
dtD 3pq.q
2 p2/ sin t cos t.p2 sin2 t C q2 cos2 t/5=2
, pi
t D 0; =2; 3=2, pi . , pi pi
pi pi pi.
3.3.3 ( )
pi pi R2 .
5... pi 1909 pi Syamadas
Mukhopadhyaya.
-
3.3. 61
pi , pi-
pi pi pi pi pi .
pi pi pi pi R2
` . pi pi, pi
pi . pi pi L pi
, pi pi Ps > 0 Ps < 0( Ps 0 Ps < 0 ). , s pi 0; `, pi pipi pi
pi p q . pi pi p q, s , ( 2.2.7),
. p q , Ps > 0 pi pi pi pi pi p
q, Ps < 0 . pi pi pi , r . pipi p; q r , pi
pi Ps > 0 Ps < 0. pi pi pi Ps > 0, pi Ps < 0 ( ). pi.
p q
a
3.5.
a L, a > 0 pi L a < 0 . pi Ps. a/ pi> 0 pi< 0, pi pi L pi. pi Z `
0
Ps. a/dt 0; (3.8)
pipi> 0 pi pipi < 0 .
pi Pns D st (. 2.2.1) piPs D .s/P s P D .s C ns/P;
pi (3.8) pi pi
.s C ns/ a, pi . `-pi
.t C `/ D .t/ t ;
pi pi t pipi pi t pi
`-pi:
t.t C `/ D P.t C `/ D P.t/ D t.t/:
-
62 3. pi
pi =2
ns.t C `/ D ns.t/; pi s.t C `/ D s.t/. .t C `/ D .t/ t , pi (3.8) Z `
0
P.t/dt D .`/ .0/ D 0:
pi, pipi .
3.3.1 3.1.2 .
3.3.2 1.1.7 (pi.
3.1.3).
3.3.3 pipi pi t D t0 ( 2.2.7) t D t0.
-
4pi
pi pi-
pi. pi pi , -
pi, pi
, pi pi pi pi -
pi. , pi,
pi pi pi
pi. pi pi pi
pi, pi pi pipi. pi
pi pi pi pi
( 5.6).
4.1 pi;
pi pi R3 pi R2
pipi , pi pi , pi,
pi , pipi pi-
pi pi pi pi .
pi , pipi pi
pi pi . pi
Rn n 1, n D 1; 2 3. , piU Rn , a U ,
pi u 2 Rn pi pi pi pi a pi U :
a 2 U ku ak < H) u 2 U:
63
-
64 4. pi
pi, Rn , pi
Dr .a/ D fu 2 Rn j ku ak < rg ; pi pi a r > 0. ( n D 1 pi , n D 2 .)
Dr.a/ D fu 2 Rn j ku ak rg , , pi
pi pi pi pi.. pi .a1 C r; a2; : : : ; an/ 2 Dr .a/ Dr.a/ (..
a1 C r C 2 ; a2; : : : ; an
).
pi, X Y pi Rn Rm, ,
pi f W X ! Y a 2 X X a pi f Y f .a/. , f
a , > 0, pi > 0
u 2 X ku ak H) kf .u/ f .a/k < : f X . -
pi .
,
: f , V Rn,
pi U Rm U \ X D fx 2 X j f .x/ 2 V g. f W X ! Y , 11 pi, pi
f 1 W Y ! X pi , f X Y .
pi pi pipi -
pi R3.
4.1.1
pi S R3 pi , p 2 S, pi pi U R2 pi W R3 pi pi p
S \W U . pi piS S \W , piW pi R3, pi S. W U ! S \W pi pi1 pi pi S \ W S. pi pi pi S
S.
4.1.2
pipi R3 pi pi pi pi
pi. , a pipi, p q
pi pi pipi . v
pipi, v a pi pipi, piv a D up C vq
1... pi surface patch pi
pi pi, pi .
-
4.1. pi; 65
pi u v. pi pi pi
.u; v/ D aC up C vq; pi
1.v/ D ..v a/ p; .v a/ q/ :pi 1 , -. ( pi pi).
pi pi pi pi-
pi pi.
4.1.3
R3 pi pi
pi (pi ) pi (pi
). pi, 1
z, pi ,
S D f.x; y; z/ 2 R3 j x2 C y2 D 1g: pi pi S :
.u; v/ D .cosu; sinu; v/: .u; v/ 2 S .u; v/ 2 R2, S . pipi, . 11 -
, .u; v/ D .uC2; v/ .u; v/. pi 11 pipi pi u 2 , 0 u < 2 . pi jV
V D f.u; v/ 2 R2 j 0 u < 2g 11, V pi R2 jV pi. pi R2 pi pi V
U D f.u; v/ 2 R2 j 0 < u < 2g; pi jU U pi., jU pi S, pi pi pi pipi pi
S x D 1, y D 0 ( 4.1). pi S,
pi. pi pi j QU , pi
QU D f.u; v/ 2 R2 j < u < g; pi pi pi S pi pipi pi S
x D 1, y D 0. S pi pi pi jU , j QU , f jU ; j QU g S S pi.
-
66 4. pi
4.1. pi x D 1, y D 0.
4.1.4
R3 pi pi pi (pi
) pi (pi ).
pi, 1 , pi
, :
S2 D .x; y; z/ 2 R3 j x2 C y2 C z2 D 1 : pi S2 pi -
pi ': p ,
pi pi p pi z pipi xy
, q. pipi , q p, '
q x. pi -
pi, pi
' .
pi pi pipi
p '. pi 0, p q
pi p sin .
p pi pi pipi xy 0, q
r , pi r pi pi q x. pi
kqk D cos , p kqk cos' D cos cos' kqk sin ' D cos sin ';
. pipi pi
p D .cos cos'; cos sin '; sin /:
-
4.1. pi; 67
x
y
z
qr
p
4.2. .
pipi , pi .; '/,
pi S2.
pi pipi , 11 ( pi)
.; '/ D .; 'C2/. pi pi
2
2; 0 ' 2:
.; '/ pi pi -
pi R2. pi pi-
pi
U Dn.; '/ j
2< 0 b c U . pi pi pi U pi pi b c, pi a. pi-
pi D pi S, pi p q, pi . (
, pi .)
pi . ( pi -
pi pi pi
.)
, , pi pi S[SC, pi pi pi pi pi W U ! R3,pi U D R2 n f.0; 0g, pi pi pi pipi xy:
.u; v/ D .u; v;pu2 C v2/:
pi pi pi , a pi S
pi pi. pi-
pi W U ! S \ W Q W QU ! S \ QW
-
70 4. pi
a
q
p
c
b
pi
4.6. pi .
pi a 2 S \ W \ QW . Q , 1.S \W \ QW / Q1.S \W \ QW / V U QV QU ,. 1 Q W QV ! V pi - pi Q . pi ,
Q . Qu; Qv/ D .. Qu; Qv// . Qu; Qv/ 2 QV .
u
U~
V~
~
V
U
4.7. pi .
4.1.1 pipi xy pi.
4.1.2 pi x W U ! R3 S2 x2 C y2 C z2 D 1 pi x y z:
x.u; v/ D .p1 u2 v2; u; v/:
pi U D .u; v/ 2 R2 j u2 C v2 < 1. pi
y
z ( U ) pi y
-
4.1. pi; 71
4.8. pi.
z, . pi pi
S2 pi ( 4.8).
4.1.3 pi
S D .x; y; z/ 2 R3 j x2 C y2 z2 D 1 : , ,
.x z/ cos D .1 y/ sin ; .x C z/ sin D .1C y/ cos pi S, pi pi
( 4.9). pi S pi pi
pi pi pi. ( pi-
pipi 4.1.3.)
S
, pi
. pipi
pi S pi .2
4.1.4 pi
pi, pi. ( pi -
pi pi.)
4.1.5 pi pi pi.
2... doubly ruled surface, pi pi pi pi-
. pi .
-
72 4. pi
4.9. pi pi.
4.2 pi
pi pi
pi ( ). pi, pipi
pi
pi . pipi pi
pi pipi .
, U pi Rm, pi-
f W U ! Rn pi n f , pi U ! R, pi . pipi pi f pi .
pi, m D 2 n D 3, f .u; v/ D .f1.u; v/; f2.u; v/; f3.u; v//;
@f
@uD@f1
@u;@f2
@u;@f3
@u
;
@f
@vD@f1
@v;@f2
@v;@f3
@v
pi . pi
:
@f
@uD fu; @f
@vD fv;
@2f
@u2D fuu;
@2f
@u@vD fuv;
@2f
@v@uD fvu;
@2f
@v2D fvv;
... pi pi pi fuv D fvu f .3
3... pi Schwarz.
-
4.2. pi 73
pi W U ! R3 . pi pipi .
4.2.1
pi W U ! R3 u v .u; v/ 2 U . , pipi u v pipi U .
pi pi pipi
4.4.
pi pi pi
.
4.2.2
S pi, pipi pi S
pi W U ! R3 pi Upi pi S. pi piS ,
p 2 S, pi pipi pi pi pipi p 2 .U /. A pipi pi pi S S A
pi S.
4.2.3
pipi 4.1.2 pi. ,
.u; v/ D aC up C vq pi u D p v D q , p q pi .
4.2.4
( 4.1.3) pi. ,
.u; v/ D .cosu; sin u; v/ pi
u D . sinu; cosu; 0/; v D .0; 0; 1/ pi .u; v/. jU j QU pi.
4.2.5
S2 4.1.4, pi
Q . D . sin cos'; sin sin '; cos /; ' D . cos sin '; cos cos'; 0/;
' D . cos2 cos'; cos2 sin '; sin cos /
-
74 4. pi
k 'k D j cos j. .; '/ 2 U , =2 < < =2, cos 0. pi pi pi Q .
4.1.2 pi pipi pi-
pi pi S2 ( pi
. 4.2.2). pi : pi pi
pipi pi ; pi -
pi pi pipi pi .
4.1.2 4.1.4
. pi pipi ( .
4.5.1) pi pi pi pi S
pi pi pi pipi pi .
pi pipi pi.
pi , pi
pi pi .
4.2.6
pi pi .
pi 5.6. pi pi
4.2.6.
4.2.7
U QU pi R2 W U ! R3 pi. W QU ! U 11 pi pi pi 1 W U ! QU , Q D pi.
pi Q pi pi-. , .u; v/ D . Qu; Qv/. ,
Q Qu D@u
@ Quu C@v
@ Quv; QQv D@u
@ Qv u C@v
@ Qv v;
pi
Q Qu QQv D@u
@ Qu@v
@ Qv @u
@ Qv@v
@ Quu v : (4.1)
pi pi pi
pi
J./ D@u@ Qu
@v@ Qu
@u@Qv
@v@Qv
. pi pi pi , Q pi R2,
J. Q / D J. Q/J./:( pi pi
pi Q pi pi Q .)
-
4.2. pi 75
D Q D 1 pi J.1/ D J./1. , J./ , , pi pi . 4.1
pipi Q . pi Q pi pipi pi-
, Q pi pipi. pipi pi-
Q , D Q 1. pi , W U ! S \ W Q W QU ! S \ QW
pipi pi pi S, V U QV QU pi pi .V / D Q . QV / D S \ W \ QW , D 1 Q W QV ! V 11 pi, , 4.2.6 . pi, Q pi pi .
pi pi pipi pi pi -
pi . , pi
pi pi
pi pi pi pi.
pi pi .
pipi pi
pi S, pi S
pipi pi S.
, pi pi pi-
pi , pi pipi S
pi pi pi S.4 pi
, pi pi
S pi, pi pi pi-
S. pi, pi S
5 (. 4.2.9). pi pi
, : pi 4.1.5
S pi pi , pi pipi pi.
4.2.1 , f .x; y/ ,
f.x; y; z/ 2 R3 j z D f .x; y/g pi pi pi pi
pi
.u; v/ D .u; v; f .u; v//:, pi pi . 5.6.4.
4... pi pi , . pi 2
. 3.5... pi .
-
76 4. pi
4.2.2 pi pi S2
4.1.2 . pi pi
pi .
4.2.3 pi pi pi ( pipi
u; v 2 R);(i) .u; v/ D .u; v; uv/,(ii) .u; v/ D .u; v2; v3/,(iii) .u; v/ D .uC u2; v; v2/.
4.2.4
x2
p2C y
2
q2C z
2
r2D 1;
pi p, q r , pi.
4.2.5 6 pipi pi C pipi
pi l pi C. pi pipi
xz, l z, a > 0 pi C pi l , b < a
C. pi pi
.; '/ D ..aC b cos / cos'; .aC b cos / sin '; b sin /:
4.10. .
4.2.6 pi pi pi
pi ( 4.11).
pi pi z, pi
.u; v/ D .v cosu; v sinu; u/;pi . pi pi -
pipi 7 p
z pi p pi z.
6... pi torus. pi.7... 4.5.
-
4.3. pi 77
4.11. pi.
4.2.7 .s/ pi R3 pi
pi. a > 0 pi pi
pi pi
.s; / D .s/C a.n.s/ cos C b.s/ sin /;pi n b .
pi pi. pi
pi pi pi a1. , , pi -
pi pi pi
pi 2a. pi :
pi .s; / pi
pi U R2, W U ! R3 11 ( ) . 5.6.3. pipi pi -
(.. 12.2.5).
4.2.8
R3 pi pi pi (. 4.12).
4.2.9 pi pi pi.
4.3 pi
pi f W S1 ! S2, pi S1 S2 pi. pi pi pi ,
pi pi pi -
.
pi pi pi pi ,
pi pi S1 S2 pi pi pi-
1 W U1 ! R3 2 W U2 ! R3, pi pi pi pi pi pi pi S1
-
78 4. pi
4.12. pi .
S2. 1 2 11 pi, pi pi f W S1 ! S2 pi-pi pi 12 f 1 W U1 ! U2, f pi ( pi -
R2 ). pi, pi Q1 W QU1 ! R3 Q2 W QU2 ! R3 pi 1 2, pi pi- 1 W QU1 ! U1 2 W QU2 ! U2, . pi pi Q12 f Q1 W QU1 ! QU2 12 f 1 W U1 ! U2 . ,
Q12 f Q1 D Q12 .2 12 / f .1 11 / Q1D . Q12 2/ .12 f 1/ .11 Q1/D 12 .12 f 1/ 1;
1 12 pi ( pi R
2). -
pipi pi pi pi
pi.
pi f W S1 ! S2 pi 11 pi pi pi f 1 W S2 ! S1 . pi ,8 S1 S2 -
pi . pi
.
4.3.1
f W S1 ! S2 . 1 pipi pi S1, f 1 pipi pi S2.
8... diffeomorphism. pi -
.
-
4.3. pi 79
pi pi pi S1 S2 pi pi pipi
pi 1 W U1 ! R3 2 W U2 ! R3, . pi f , f .1.u; v// D 2.F.u; v//, pi F W U1 ! U2 11 pi , F1 . pi pipi pi 4.2.6.
pi pi-
pi pi . pi f WS1 ! S2 pi pi , p 2 S1, pi piO S2 f .O/ pi S2 f jO W O ! f .O/ (pi pi pi pi . 4.2.9). pi
pi (piO D S1). pipi, 4.3.1 f pi , pi pipi pi
f 1 11.
4.3.2
pi pi pipi yz pi S pi
pi y pi z
pi pi pipi xy. pi pi
f .0; y; z/ D .cosy sin y; z/:
4.13. .
, f 11 pipi -
pi pi S. f pi -
pi pi pi pipi yz -
pi .u; v/ D .0; u; v/ pi f jU ; j QU g S 4.1.3. p D .0; a; b/ pipi yz. a pipi 2 , pi n
2n < a < 2.nC 1/ , f ..u; v// D .u 2n; v/ 2n < u < 2.nC 1/;
-
80 4. pi
pi pi f pi pi
O D f.0; y; z/ j 2n < y < 2.nC 1/g pipi pi
f .O/ D f.x; y; z/ 2 S j x 1g S. a pi pipi ,
j QU jU ( pi ). a pi pi pipi , pi ,
p pipi, pi pi O pipi pi
pi p pi f pi pi
S.
4.3.1 S pi, S ! R. , S pi, pi-
S ! R3 S ! R.
4.3.2 S x2 C y2 D z2 z > 0 (. 4.1.5). pi f pi pipi f.0; y; z/ j y > 0g S f .0; y; z/ D .y cos z; y sin z; y/. f pi .
4.4 pi pi
pi pi S () pi
S. pipi pi
pi.
4.4.1
pi pi S p 2 S pi p pi S pi pi p. pi TpS
S p pi S p.
pi TpS, pi pi-
W U ! R3 S p , .u0; v0/ D p. pi S pi p pi..t D t0, pi u.t/ v.t/
.t/ D .u.t/; v.t// (4.2) t t0, u.t0/ D u0, v.t0/ D v0. u v ( pi 5.6). ,
pi t 7! .u.t/; v.t// , . 4.2 pi pi S.
-
4.4. pi pi 81
4.4.2
W U ! R3 pi pi S pi pi p 2 S, .u; v/ U . pi S p - pi R3 pi pi pi u v (
pi pi .u0; v0/ 2 U pi .u0; v0/ D p).
pi pi S,
.t/ D .u.t/; v.t//: d=dt , pi
P D u PuC v Pv:pi P u v . ( - pi : u v u v pi u v
u.t/ v.t/ , pi
t ,
d=dt .)
, pi R3 pi pi
pi u v uCv pi .
.t/ D .u0 C t; v0 C t/ pi S, t D 0, p 2 S,
P D u C v : u v
pi p pi pi S.
pi , u v .
pi, pi pi
pipi. pi 4.4.1 pipi pi pipi
pi pi pi pi p, pi -
pi pi 4.4.2 (. 4.4.2). pi
u v pi pi pipi
pi .u0; v0/ pi pi pi-
pi pi, pi u 7! .u; v0/ v 7! .u0; v/( pi pi pi pi u D u0 v D v0).
pi pi pipi pi,
pi pi pi pi.
pi pi f W S ! QS pi. pi f p 2 S pipi pi pi f .p/ 2 QS p pi S, pi q. p
q pi , pi pi
pi S p. pi f p pipi
pi S p pi
-
82 4. pi
QS f .p/. , pi f p pipi piDpf W TpS ! Tf.p/ QS .
Dpf , w 2 TpS pi S p. , w pi
p pi S pi pi p, w D P.t0/. pi Q D f pi QS pi pi f .p/ t D t0, Qw D PQ.t0/ 2 Tf.p/ QS .
4.4.3
pipi, piDpf f p 2 S piDpf W TpS ! Tf.p/ QS piDpf .w/ D Qw pi w 2 TpS.
pi pi pi pipi
, Dpf .w/ pi f , p w:
pi pi (pi pi) pi pi w
p a priori Dpf .w/ pi pi pi pi.
W U ! R3 pi S pi pi p, p D .u0; v0/, ; U pi
f . .u; v// D Q ..u; v/; .u; v//:w D uCv pi p pi.t/ D .u.t/; v.t//,pi u v Pu.t0/ D Pv.t0/ D . pi QS Q.t/ D Q . Qu.t/; Qv.t//, pi Qu.t/ D .u.t/; v.t// Qv.t/ D .u.t/; v.t//,
Dpf .w/ D PQu Q Qu C PQvQv D . Puu C Pvv/ Q Qu C . Puu C Pvv/ QQv;pi pi u v pi t0. pi,
Dpf .w/ D .u C v/ Q Qu C .u C v/ QQv : (4.3) pi p, f , , pi p, f w, pi
pi.
pi . 4.3 pipi pi pi pi.
4.4.4
f W S ! QS pi pi p 2 S, piDpf W TpS ! Tf.p/ QS pi.
. 4.3 pi pi Dpf pi
fu; vg Tpf f Q Qu; QQvg Tf.p/ QS piu vu v
pi .u; v/ 7! ..u; v/; .u; v//.
-
4.4. pi pi 83
4.4.5
(i) S pi p 2 S, pi p pi- S ! S pi TpS ! TpS.
(ii) S1;S2 S3 pi f1 W S1 ! S2 f2 W S2 ! S3 pi, p 2 S1,
Dp.f2 f1/ D Df1.p/f2 Dpf1:
(iii) f W S1 ! S2 , p 2 S1 piDpf W TpS1 ! Tf.p/S2 .
pi (i) pi. (ii), w 2 TpS1 pi p pi 1 S1. pi 2 D f1 1 pi S2 pi Dpf1.w/ f1.p/, 3 D f2 2 D .f2 f1/ 1 pi S3 pi Df1.p/f2.Dpf1.w// f2.f1.p//.
pi 3 p pi Dp.f2 f1/.w/., (iii), g W S2 ! S1 pi f ,
g f f g pi S1 ! S1 S2 ! S2, .pi (i) (ii) pipi Df.p/g pi
Dpf .
pi pi pi pi pi pi-
pi .
4.4.6
S QS pi f W S ! QS pi, f - pi , p 2 S, piDpf W TpS ! Tf.p/ QS .
pi pi f pi p 2 S. pipi pi pi O S pi pi p -
f .O/ pi QS f jO W O ! f .O/ . 4.4.5(iii), Dpf (pi-
pi TpS D TpO). pi pi
pi 5.6.
4.4.1 pi pipi pi pi-
pi :
(i) .u; v/ D .u; v; u2 v2/, .1; 1; 0/.(ii) .r; / D .r cosh ; r sinh ; r2/, .1; 0; 1/.
-
84 4. pi
4.4.2 , .u; v/ pi, -
u v pi pi
.
4.4.3 S pi, p 2 S F W R3 ! R . rSF pi rF D .Fx ; Fy ; Fz/ F pi TpS. , pi S pi pi p
t D t0,
.rSF / P.t0/ D ddt
tDt0
F..t//:
pi rSF D 0 pi F S pi pi p.
4.4.4 f W S1 ! S2 pi pi S1. f pi S2.
4.5 pi
pi pipi TpS pi p 2 S pi R3, pi pipi pi
, pi
S p. , pi ,
4.4.2, pi pi W U ! R3 pi pi p, pi,
N D u vku vk(4.4)
(pi pi pi U pi p),
pi u
v . N pipi 9 -
pi p. pi pipi, N
pi pi pi pi p. , Q W QU ! R3 pi pi pi p, pi
4.2.7
Q Qu QQv D det.J.//u v;pi J./ pi pi pi Q .pi pipi Q
N Q DQ Qu QQv
k Q Qu QQvkD u vku vk
D N ;
pi pi pi J./.
pi .
9... standard unit normal.
-
4.5. pi 85
4.5.1
pi S pi pi A S
: pi pipi pi-
A, det.J.// > 0 pi .
pi pi pi pi:
4.5.2
S pi pi A pi
4.5.1, pi pi
S: pi pipi pipi -
pi A.
pi pi pi S pi
N , piN WS!R3( pi N
S ! R) , p 2 S, N .p/ - TpS. pi pi pi! pi-
pi S pi
.u; v/ uv pipi N S, pipi. pi pi pi
pi pi 4.5.1. pi
pi ( pi pi pi-
pi pi ). pi , pi pi-
pi S, pi pi S pi
pipi pi
S.
pi pi pi pi pi. -
pi pi .
4.5.3
Mobius pi pi pi
l pi p, p pi C
pi, p pi pi C, l
pi pi p. pi C x2 C y2 D 1 pipi xy, l 1 pi pi
z p D .1; 0; 0/, p pi pi z, l pipi pi =2 pi
p pipi pi pi p z. l pi
.1; 0; t/
.t; / D
1 t sin 2
cos ;
1 t sin
2
sin ; t cos
2
:
pi
U D f.t; / 2 R2 j 1=2 < t < 1=2; 0 < < 2g:pi pi Q pi pi
-
86 4. pi
pi pi QU D f.t; / 2 R2 j 1=2 < t < 1=2; < < g: pi pi pi, pi - Mobius pi pi pi
pi. Mobius pi-
S (. 4.5.1).
4.14. Mobius.
pi pipi N
(pi t D 0).
t D sin
2cos ; sin
2sin ; cos
2
; D . sin ; cos ; 0/;
pi
t D cos cos
2; sin cos
2; sin
2
:
, N .
Mobius pi pi -
N S pi pi
pi S. pipi , .0; / ,
N D ./N ;pi W .0; 2/! R ./ D 1 . pi ./ D C1 2 .0; 2/, ./ D 1 2 .0; 2/. N N pi, pi pi D 1. .0; 0/ D .0; 2/ ( N )
N D lim#0
N D .1; 0; 0/ pi N D lim"2
N D .1; 0; 0/:
pi, Mobius pi.
pi S pi, pi-
pi S. pi
pipi .
p 2 S N pi p. pi pi pi pipi TpS
-
4.5. pi 87
, pi -
pi TpS,
N . , pi N pipi pi
TpS: pi pi pi N . pi,
TpS pi pi, pi
pi TpS .
TpS
N
4.15. pi.
v w TpS, pi
( pi pi pi ) v w
pi pipi pi v , pi
pi pipi w.
cvw. cwv D cvw;
pi cvw pi - TpS. pi cvw pipi pipi 2 .
4.5.4
p 2 S2 pi pipi p. p ,
p p, , p, -. pi pi S2: pi
pi .
pi, pi pi , pi
p .1; 0; 0/ , pi
pi v D .0; 1; 0/ w D .0; 0; 1/ cvw D =2( 3=2; 7=2 pi.). pi ,
pi pi.
-
88 4. pi
4.5.1 pi pi pi-
Mobius 4.5.3. pi
R2,
pi C1 1 .
4.5.2 pi pi S QS S pi. pi QS pi.
-
3pi pi
1.1.6 : .t/ D .p cos t; q sin t/. (i) - pi p.1 cos t/. (ii) n D .q cos t; p sin t/ pi .t/ pi
j.p f1/ nj=knk j.p f2/ nj=knk.1.2.3 pi 3.2.2.
1.4.5 (ii) (i) pi T1 0 T1-pi. fTr T1g. (iii) pi .
1.4.6 .t/ D .1.t/; : : : ; n.t//. i T -pi, pi, 1.4.5, i , () pi, Ti .
1.4.4, i , T D kiTi pi ki . k pi pi ki .
pi T0 D T=k.1.5.2 1.5.1 f
.@f@x;@f@y;@f@z/ pi f .x; y; z/ D 0 pi
pi pipi . pi
5.6.
1.5.3 .
2.2.3 s pi pi pi .
2.2.10 pi .s/ pi
. pi s, pi .s/
p C sa `, pi .s/ `. pi .s/. P.s// D a. :(i) S n n pi ( S
S t S) v n 1 pi, vtSv D 0.
373
-
374 3. pi pi
(ii) A pi ( AtA D I ) - pi s, At dA
ds ( dA=ds
pi A).
2.3.2 pi pi .
2.3.3 pi a D t cos C b sin .2.3.4 a r , .a/.a/ D r2
pi pi. C nC Pb.2.3.6 pi pi pi pi -
vi vj pi .
3.2.1 pi pi 1.
3.3.1 pi 2x1x2 x21 C x22 .4.1.4 pi pi U .
5.2.3 pi pi : .t/ D aC tb. pi- . 5.1 pipi t .
pi pi
pi pi .
5.6.1 pi 4.2.6.
5.6.4 .u; v/ D .f .u; v/; g.u; v/; h.u; v//, pifu fvgu gv
-
. pi 4.2.6 pi V
W F1 W V ! W F1.f .u; v/; g.u; v// D .u; v/ .u; v/ 2 W . '.x; y/ D h.F1.x; y//.
6.1.5 pi pi pi u D u0 R v1v0
pG.u0; v/dv.
6.3.8 pi .2.5.
6.4.3 pi pi
.
6.5.3 pi pipi pi p pi.
6.5.4 Mobius pi (i) b 2 R.pi Mobius -
Mobius.
7.1.2 pi .u N /u, pi Nu Nv u v , pi N
.
7.3.4 pi 6.4.7.
7.4.3 pi 7.4.7.
8.1.6 2 2 pi
A Da c
b d
pi A2 .aC d/AC .ad bc/I D 0.
8.1.8 pi 8.1.6.
8.2.3 pipi pi EN D GL EN GL.
-
3. pi pi 375
8.2.5 pi 8.2.2.
8.2.6 (i) . 8.5.
8.2.7 pi 8.1.2, pi 8.2.1
8.1.1.
8.2.8 pi 6.1.4 7.1.3.
8.3.1 (iii) pi pi .
8.4.1 pi 8.2.4 7.3.3 8.4.3.
8.5.2 pi 8.1.1.
8.5.3 pi 8.2.4 pi 8.1.2.
9.1.1 pi 9.1.4 9.1.6.
9.1.3 pi 4.2.7.
9.1.4
x2
p2C y
2
q2C z
2
r2D 1
pi .t/ D .f .t/; g.t/; h.t//
. Rf ; Rg; Rh/ D f
p2;g
q2;h
r2
pi .t/.
9.1.5 pi 8.2.2.
9.2.2 pi 6.2.1.
9.3.3 pi , pi w > 1,
v pi pi . 9.14 pipi 2 .
9.4.2 S2 pipi pi pi p q.
9.4.3 (i) pi LHospital.
9.5.2 (ii) , pipi p
R, q0. pi pi p q0.
10.1.1 pi pi piWeingarten.
10.1.3 P D Lw2.10.2.2 pi 6.1.4 9.5.1.
10.2.4 pi , 9.5.1 10.2.3(ii).
10.3.1 pi pi 10.3.4, 1 pi
p pi, 2 D 2H 1 pi .
10.4.2 pi
2 0
0 1
.
10.4.3 pi 9.2.1.
11.1.4 pi pi 11.1.4 .2.3. pi-
2R pi pi i.b C r/ i.b r/, pi.11.2.1 pipi pi a b .
11.2.2 pi 11.1.1.
-
376 3. pi pi
11.2.3 pi pipi pi l -
pi 11.1.3.
11.2.4 (i) P H, pi pi pi
P l m. (iii) l m pi 11.2.3,
F H. pi pi 11.2.3 pipi
pi G pi pi F.i/ i
F.l/ l . pi pi (ii).
11.2.5 pi .2.2.
11.3.2 pi 11.2.4.
11.3.3 pi 11.2.5.
11.3.4 pi 11.2.5.
11.3.6 pi pi
pi. pi pipi
pi pi a pi pi
b c pipi pi
b c, pipi pi .
11.3.8 pi pipi pi -
11.3.7 pi pi 11.3.6.
11.4.2 H pi l .
11.4.3 DP pi a D 0, b 2 R.11.5.3 pi pi M.z/ D .a; bI c; z/ Mobius
pi pi .a; b; c/ .1; 0; 1/. pi M Mobius pi pi .a; b; c/ .1; 0; 1/, M.z/ D .1; 0I 1;M.z// D .a; bI c; z/.
12.1.1 pi 8.2.9.
12.1.3 pi 8.6.1 12.1.1.
12.2.4 pi 8.5.1.
12.3.1 pi 8.1.6 8.2.9.
12.5.1 pi D e Q .12.5.3 (ii) pi pi Weierstrass . 12.25 12.26.
13.1.2 pi 13.1.2.
13.3.2 k.t/ D .nt k/, pi pi 8.5.3(iii) 'k D k=. 1 C C n1/.
13.5.2 E 12V.V 1/.13.7.1 13.7.2 pi
de Moivre.
13.7.3 Q pi Q , d=ds D .1 2/1=2 P, pi D cos . pi pi Green
R `./0 .d=ds/ds D 0.
13.8.1 pi ( pi )
vtHv < 0 ( > 0) 2 1 pi v.
-
4
1
1.1.1 pi pi x 0.1.1.2 (i) .t/ D .sec t; tan t/ =2 < t < =2 =2 < t < 3=2.
pi :
pi pi pi , y 1 y 1.(ii) .t/ D .2 cos t; 3 sin t/.
1.1.3 (i) x C y D 1. (ii) y D .ln x/2.1.1.4 (i) P.t/ D sin 2t.1; 1/. (ii) P.t/ D .et ; 2t/.
4.1.
377
-
378 4.
1.1.5 P.t/ D 3 sin t cos t. cos t; sin t/ sin t D 0 cos t D 0, t D n=2 pi n . ( 4.1) (. 1.3.3.)
1.1.6 (i) pi p pi .p cos t p/2 C q2 sin2 tD .p2q2/ cos2 t2p2 cos tCp2 D p2.1 cos t/2, pi 2p.
(ii) P D .p sin t; q cos t/, n D .q cos t; p sin t/ n P D 0. pi pi pi .t/
.p cos t p; q sin t/ nknk D
pq.1 cos t/.p2 sin2 t C q2 cos2 t/1=2
p2q2.1 2 cos2 t/.p2 sin2 t C q2 cos2 t/ D q
2:
(iii)
.p f1/ nkp f1k
D .p f2/ nkp f2k:
pi pipi q.
1.1.7 pi t , .at; a/, pi
pi .0; 0/ pi
.a.t sin t/; a.1 cos t// ( 4.2).
ta
at0
4.2.
1.1.8 pi .x; y; z/ x2 C y2 D 1=4 -
x C 12
2C y2 C z2 D 1. pi 1 z 1, pi
z D sin t . pi xC 14 C sin2 t D 34 , x D12sin2 t D cos2 t 12 . pi pipi pi pi y D sin t cos t (y D sin t cos t , pipi pi t 7! t).
1.1.9 P D .2 sin t C 2 sin 2t; 2 cos t 2 cos 2t/ D p2.p2 1; 1/ t D =4. pi y . 1p
2 1/ D .x p2/=.p2 1/
y . 1p2 1/ D .x p2/.p2 1/.
1.2.1 P.t/ D .1; sinh t/, k Pk D cosh t s D R t0 cosh udu D sinh t .
-
4. 379
1.2.2 (i) k Pk2 D 14 .1C t/C 14 .1 t/C 12 D 1. (ii) k Pk2 D 1625 sin2 t C cos2 t C 925 sin2 t Dcos2 t C sin2 t D 1.
1.2.3 d=d , P D . Pr cos r sin ; Pr sin C r cos /, k Pk2 D Pr2 C r2. pi r D Pr D 0 pi . Pr2 D 1 r2, pi pi pipi r D sin.C/ pi . pi 1=2, 3.3.
1.2.4 u , j P uj D k Pk cos , pi P u, P u k Pk. pi
.q p/ u D ..b/ .a// u DZ ba
P udt Z bak Pkdt:
u D .q p/=kq pk pi.1.3.1 (i) P D sin 2t.1; 1/ t pipi =2,
. (ii) , P 0 0 < t < =2. (iii) P D .1; sinh t/ pi pi , .
1.3.2 x D r cos D sin2 y D r sin D sin2 tan , pi - 7! .sin2 ; sin2 tan /. 7! sin 11 pi pi pi .=2; =2/ .1; 1/, t 7! sin1 t , t D sin pi pi. sin2 D t2, sin2 tan D t3=
p1 t2, pi pi pi
.
1.3.3 (i) P D 0 t D 0() m n 2. m > 3, pi R 0 t D 0, R R t D 0 pi n > 3. pi pi pipi: .m; n/ D .2; 2/ .3; 3/ R t D 0, pipi pi pi .m; n/ D .2; 3/ .3; 2/ pipi pi R t D 0. (ii)pi pi .t/ D
t2; t
3p1t2
pi P D 0,
R D .2; 0/ D .0; 6/ t D 0, . (iii) Q.Qt/ pi .t/ t D t0, t D t0
d Q=d Qt D .d=dt/.dt=d Qt / D 0;d2 Q=d Qt2 D .d2=dt2/.dt=d Qt /2 d3 Q=d Qt3 D .d3=dt3/.dt=d Qt /3 C 3.d2=dt2/.dt=d Qt /.d2t=d Qt2/:
pi dt=d Qt 0, pi d2 Q=d Qt2 d3 Q=d Qt3 t D t0.
1.3.4 (i) Q.t/ D .'.t//, pi pi '. pi Q. .t// D .'. .t/// D .t/. (ii) Q.t/ D .'.t// O.t/ D Q. .t//, pi' pi pi, O.t/ D ..' /.t//. ' pi pi d
dt.'. .t// D P'. .t// P .t/ 0,
P' P 0.1.4.1 .t C 2/ D .t/ t . pi .t/ D .u/,
cos3 t.cos 3t; sin 3t/D cos3 u.cos 3u; sin 3u/. cos3 t D cos3 u, cos t D cosu, u D t; t; C t 2 t ( pi pipipi 2). pipi
-
380 4.
t D n=3 pi n, t . pi pi , t D =3; 2=3, pi pi pi u D 2=3; =3, . pi
.=3/ D .1=8; 0/.
1.4.2 pi Q.t/ D .cos.t3 C t/; sin.t3 C t// pi
.t/ D .cos t; sin t/ .
1.4.3 T -pi, kT -pi k 0 ( pi pi k k > 0, k k < 0). T1-pi T2-pi, k1T1-pi k2T2-pi
k1; k2, .tCk1T1Ck2T2/ D .tCk1T1/ k2T2-pi, pi .t/ k1T1-pi.
1.4.4 T -pi. T D kT0 C T1, pi k 0 T1 < T0. 1.4.3 T1-pi T1 > 0 pi T0.
1.4.5 (i) pi T1 > 0 T1-pi, T1 -
, pi T2 > 0 -
T2-pi. pi -
. (ii) fTr gr1 (i) , pipi pi T1 0. pi T1-pi ( ) .t C T1/ D limr!1 .t C Tr / Dlimr!1 .t/ D .t/. 1.4.3, .Tr T1/-pi r 1, 0. (iii) fTr g (i) Tr ! 0 r !1, pi pi0 D .f .t C Tr / f .t//=Tr D Pf .t C Tr / pi 0 < < 1. r !1pi Pf .t/ D 0 t , f .
1.4.6 pi, T0 D .ki=k/Ti pipi Ti , i T0-pi. T pipi pi
fTi ; 2Ti ; : : : ; kiTi g i pi i , P D fT 0 2 T j T 0-pig. P pipi ( T pipi-) ( T 2 P). P T 00 pi T
00-pi ( T
0-pi, T 0 > T T 0 2 P). 1.4.4, T0 D k0T 00 pi k0, pi pi k0i T
00 D k0iTi i pi
i . pi kiTi=k D k0kiTi , kk0 ki . k , k0 D 1, T0 D T 00.
1.5.1 x.1 x2/ 0() x 1 0 x 1, pi pi pi () . pi t 1 0 t 1, pi pi y 0.
1.5.2 .t/ D .x.t/; y.t/; z.t/ pi pi f .x; y; z/ D 0, pi- f .x.t/; y.t/; z.t// D 0 pi t pi Pxfx C Pyfy C Pzfz D 0, P rf .fx; fy ; fz/. pi pi,
rf pi. pi f D 0 g D 0 pi rf rg pi .
1.5.3 .t/ D .u.t/; v.t/; w.t// pi R3. pi Pu, Pv Pw t . pi Pu.t0/ 0 x0 D u.t0/. pi pi 1.5.2, pi h.x/ x x0 t D h.x/ x D u.t/ t t0. pi , t t0, .t/ pi
-
4. 381
pi f .x; y; z/ D g.x; y; z/ D 0, pi f .x; y; z/ D y v.h.x// g.x; y; z/ D zw.h.x//. f g pi pi- , rf D .Pvh0; 1; 0/ rg D . Pwh0; 0; 1/, pi d=dx.
2
2.1.1 (i) ( 1.2.2(i)), D k Rk D k.14 .1 C t/1=2;14 .1 t/1=2; 0/k D 1p8.1t2/ . (ii) ( 1.2.2(ii)),
D k Rk D k.45 cos t; sin t; 35 cos t/k D 1. (iii)pi 2.1.2 D k.1;sinh t;0/.0;cosh t;0/kk.1;sinh t;0/k3 D
cosh t
cosh3 tD sech2t .
(iv) .3 cos2 t sin t; 3 sin2 t cos t; 0/.3 cos2 tC6 cos t sin2 t; 6 sin t cos2 t3 sin3 t; 0/ D.0; 0;9 sin2 t cos2 t/, D k.0;0;9 sin2 t cos2 t/kk.3 cos2 t sin t;3 sin2 t cos t;0/k3 D
13j sin t cos t j . pi-
t pipi =2, .1; 0/ .0;1/ .
2.1.2 pi 1.3.5, v.t/ ()
t , kv.t/k () t pi pipi v.t/ t . pi pi pi
pi 2.1.2. pi pi .t/ D .t; t3/ .t/ D 6jt j=.1C 9t4/3=2, pi pi t D 0.
2.2.1 t ns D 0 pi Pt D sns .2.2.2 , t D P Pt ns (
ns pipi pi t). pi s D Pt ns .2.2.3 pi , pi 2,
Qs D s M pi l . : pi '0 2.2.1 l
x, ( pi ) Q' D '. , Q pi, pi pi
pi pi. pi pipi,
2.3.5, pi ,
pi pi pipi pi pi
pi, pi . pi -
pi .
2.2.4 pi pi s pi-
pi a ' pi .
s s C s. - ( pi ) .T cos '/, pi
pipi , T cos ' pipi , . -
.T sin '/, pipi
, pi pipi s. pi T sin ' D s C pi ; , pipi ' D s D 0 - C. pi T cos ' D T sin ' D s pi tan' D s=a,pi a D =. pi sec2' d'
dsD 1=a, pi pi
-
382 4.
s D d'=ds D 1=asec2' D 1=a.1C tan2 '/ D 1a .1C s2=a2/1. pi pipi pi 2.2.4.
2.2.5 d=dt D d=dt C dns=dt D .1 s/ds=dt t, s
pi ds=dt D j1 sjds=dt . pi
t D .d=dt/.ds=dt/ D t, pi n D ns . pi, pi pi s pi s n
s D d t=ds D .d t=dt/=j1 sj.ds=dt/ D j1 sj1 d t=ds
D s.1 s/1ns D s.1 s/1ns D sj1 sj1ns :2.2.6 .s0/ pi .s0/
.s0 C s/. pi
..s0/1
2..s0/C .s0 C s/// ..s0 C s/ .s0// D 0:
pi Taylor pipi pi pi
s , pi (pi pi-
s0) . / Ps C 12 . R 1 R/.s/2 D 0. pipi pi s s pi pi . / P D 0 . / R D 1. pi pi pipi D C ns pi , , pi R D sns , pi pipi D 1=s .
2.2.7 pi tC 1s .st/Ps2sns D Ps
2sns , pi
u D R kPkds D R Ps2sds D u0 1s , pi u0 . pi
pi ns pi t. dns=du D st=.du=ds/ D
3s
Ps t, pi-
pi 3s = Ps .pipi .s/ .s/C ns.s/ pi . pi .s/, pi D 1=s.s/, pi pi pi pi
ns.s/, pi .s/.
d=dt , 0 D a.1cos t; sin t/, pi s pi ds=dt D 2a sin.t=2/, t D d=ds D .sin.t=2/; cos.t=2//.pi ns D . cos.t=2/; sin.t=2// Pt D .d t=dt/=.ds=dt/ D 14a sin.t=2/ .cos.t=2/; sin.t=2// D 1=4a sin.t=2/ns , pi pi 1=4a sin.t=2/
.t/ D a.t sin t; 1 cos t/ 4a sin.t=2/. cos.t=2/; sin.t=2//D a.t C sin t;1C cos t/:
pi Qt D Ct , pi a.Qtsin Qt ; 1cos Qt/C a.;2/, pipi pi pi
a.;2/.2.2.8 pi .s/ ` s,
pi .s/. pi P PC.` s/ R D s.` s/ns ( d=ds). v pi dv=ds D s.` s/, pi pi ns pi t. dns=dv D 1s.`s/ Pns D
1`s t, pi pi 1=.` s/.
-
4. 383
2.2.9 pi Q.s/ D .sinh1 s;p1C s2/. -
.s/ D Q.s/s PQ.s/ Dsinh1 s sp
1Cs2 ;1p1Cs2
D .utanh u; sechu/
u D sinh1 s. pi, .x; y/ , u Dcosh1.1=y/ x D cosh1.1=y/
p1 y2.
2.2.10 pi .s/ pi pi .s/ ` pi q .s/ pi pi .s/
pCsa. pi .s/.pCsa/ D .s/.q.s//, pi pi pipi . P .s/ D aC
dds.s/
.q.s//.s/ P.s/.
pi pi a , -
. pi pi . dds.s//.q.s//.s/.q.s// D 0.
A D .s/, v D q .s/, pipi ( pi ) .Av/t dA
dsv D 0, vtAt dA
dsv D 0. A , pipi pi
(i) (ii) pi. pi (i) pi :
vtSv D Pi;j vivjSij D Pi;j vj viSj i D Pi;j vivjSij . (ii) pi AtA D I .
2.3.1 (i) t D .12 .1 C t/1=2;12 .1 t/1=2; 1p2 // ,
Pt D .14 .1Ct/1=2; 14 .1t/1=2; 0/, D kPtk D 1=p8.1 t2/
n D 1 Pt D 1p2 ..1 t/1=2; .1C t/1=2; 0/ b D t n D .12 .1C t/1=2; 12 .1 t/1=2; 1p2 //
Pb D .14 .1C t/1=2;14 .1 t/1=2; 0//, D 1=p8.1 t2/.
Pn D t C n pi .(ii) t D .45 sin t; cos t; 35 sin t/ , Pt D .45 cos t; sin t; 35 cos t/, D kPtk D 1 n D 1 Pt D .45 cos t; sin t;35 cos t/ b D t n D .35 ; 0; 45 //, Pb D 0 D 0. pi 2.3.5, 1= D 1 C 1 n D .0; 1; 0/ pipi pi pi .0; 1; 0/ b D .35 ; 0;45 /, pipi 3x C 4z D 0.
2.3.2 a D =.2 C 2/, b D =.2 C 2/. 2.1.3 2.3.2, pi a b pi a=.a2 C b2/ D b=.a2 C b2/ D . 2.3.6, pi pi pipi .
2.3.3 t a (= ) pi n a D 0 t, n b pi- R3, a D t cos Cb pi a , D sin pi a D t cos b sin pi D cot . , D , pi D cot pipipi a D t cos C b sin , pi t a D cos , t a. 2.1.3,
pi d=d D .a sin ; a cos ; b/ z cos1.b=
pa2 C b2/.
2.3.4 pi .a/ .a/ D r2 pi t .a/ D 0t tC n . a/ D 0, pi n . a/ D 1= n tC .t C b/ . a/ D P=2,pi b . a/ D P=2 b t n . a/ D . P=2/P = D . P=2/P. , . (2.22), D . P/P, pi
-
384 4.
.2 C . P/2/P D 2 P C 2. P/. P/P D 0, 2 C . P/2 , r2 (pi r > 0). a D C n C Pb . (2.20) pipi a D t C Pn C .t C b/ C . P/PC . P/.n/ D 0, a - k ak2 D 2 C . P/2 D r2. pi pi a r .
2.3.5 P D P P, T D P t k P k2 D .P P/ .P P/ D P P P -. pi R D P R pi pi pi , pi N D Pn. pi B D P t Pn. P ( pi), P.t n/ D Pb, P , P t Pn D Pb ( .1.6).
2.3.6 ij D vi vj . v1, v2 v3 ij D ij (D 1 i D j D 0 i j ).1 pi ij D ij s, s D s0. vi vjpi ij D
P3kD1.aikkj C ajkik/. ij D ij
aij Caj i D 0. , pi
s D s0.
3
3.1.1 P D . sin t a sin 2t; cos t C a cos 2t/, k Pk2 D 1 C a2 C 2a cos t . 1 C a2 2jaj D .1 jaj/2, jaj 1. jaj D 1 k Pk D 2.1C a cos t/, . a D 0
. 0 < jaj < 1, .t1/ D .t2/ H) 1 C a cos t1 D 1 C a cos t2H) cos t1 D cos t2 H) t2 D t1 2 t1. pipi .t2/ D..1Ca cos t1/ cos t1;.1Ca cos t1/ sin t1/, .t1/ D .t2/ H) sin t1 D 0H) t1 D 0 . pipi t2t1 pipi 2 , pi pi 2 . jaj > 1, pi cos t D 1=a, pi 0 t < 2 , t1 < t2, . pi pi 1.1.7 (pi
pipi a D 2), pi - pi pi pi pi t1 t t2, .
3.2.1 1, M R2 M.v/ D P vCb,pi P 3 3 pi b . Q DM./, PQ D P P, k PQk D k Pk, pi Q . - pi pipi xy R3, . 3.2 pi
: A./ D R T0 . P R/ k dt , pi k D .0; 0; 1/. pi .1.6 pi
Q pi (pi M , pi, pi-
pi Q pi pi).1... ij Kronecker, pi :
ij D1 i D j0 i j :
-
4. 385
3.2.2 pi pi .t/ D .p cos t; q sin t/, 0 t 2 ,
R 20
qpq sin2 t C pq cos2 tdt D 2ppq.
pi , ` pi
` p4 pq D 2ppq, , p D q. R 20 k Pkdt DR 20
qp2 sin2 t C q2 cos2 tdt:
3.3.1 .x1; y1/ .x2; y2/ , xip2
C yiq2
< 1 i D 1; 2. pi .tx1 C .1 t/x2; ty1 C .1 t/y2/ pi 0 t 1.
.tx1 C .1 t/x2/2p2
C .ty1 C .1 t/y2/2
q2
D t2 x21p2
C y21
q2
!C .1 t/2
x22p2
C y22
q2
!C 2.1 t/
x1x2
p2C y1y2
q2
< t2 C .1 t/2 C t.1 t/ x21p2
C y21
q2C x
22
p2C y
22
q2
!