"!# $%&'

()* * +-,*/.10 324!#657

89$%&:5 ) */) ) */; ! 0=

)* 2 *+@?

$% > $%# !# # A B835C! 0=

ij ut]%=Puuuub:]'I]M>A'h(uuPPubbxPut]bPCn9]P3'xb|ubbNOI*P(PQRG*SAI@CTO@EUGA@EUOVWMX@IZY[

Cn Pbu6x3uZ]uhbuxZ'(b

C06O'6OPZ'

Cn, 1 n u'hbP]uC'PPuub3]bP

Cn, n 1,x3uPbPub:u:>:'PPuu]bP

CCn uOL>mPbP'hbPu u:(umx3PO|'X'uuPuu>_]bu

C0 uuu]'uuPuu>A3u'Pn > 10

](xuuu-tPmPuOxtbbxu('PPuuu]

\ bA]_^a`bJWabfxabRcnURJ_LOSy`'`aGL(n,R)

d{bid'NuRVdQC^'JLPJJHS'abfxT'NPQhjc3'd~P `e^'J]\|NPVPR{J|{VONvQabRS'T'

nn {bi J]\|NP^'Jp{p`S'JVPRSa^'VuR{J Rn2 n`'JVOWJbicWypi FmHyF QaRS'T'y~QCJ\_^'J%f]ab`'T'VxaRcG{VuRS'JWy l i T'^'^gn2jVi J{TN

34.5 X 6= 0 fxa l aNuR{`'S'JVPRS'a^'VPR{N{VPNuvQabRS'T'JHbi/abVPRcj(aW~WaW_NPVui T_J`'S'Nrl Nui TRNuic^'NPWJRJSJKqQabRST''gX

^'NpSab{NO^q^y]i FnRJTeJ`'S'N l NuiTRNi Tz{VONuvl JVPRaRJU'^JGHi TfxWTv|W

XQabRS'TpRJ]\|N6^'N(Sa{^'gM^yiFnbRN6{QCNPVPRNhV(Wa\ l JK

RJU'WJK`'S'JVuRSab^'VPR{JGL(n,R)

VOJ l NPS\TR_^'NPWJbRJSyF JW'S'NPVPR^JVPRc'jfaW'T'QJHSabfOJQhn`'Jbi ^ad1i T^'NOK'^'abdMLOSy``ad'{bi'd'NPRVdMJHbi aVPRcF {

Rn2naVui N l J{ar

RNuic^'JTQC^'JLPJJHS'abfxT'NPQhjGid l JWaf]abRNui cVuR{aRJLPJ'nhURJRJTiT T'^JN[`'J l QC^'J]\|NOVuR{JNO{Wi T l Jr

{ae`'S'JVuRSab^'VPR{a1d{bid'NuRVdQC^'JLPJJHS'abfxT'NPQ U'aVPRJ1TVO`'JbicfOyFmRg&~O/[h@ $#b %ikjhalj

maVOVPQCJRS'TQ {`'S'JVPRS'a^'VPR{NRn

VWJJS l T^abRaQCTx1, . . . , xn

QC^J\pNOVPR{JX Rn nVOJVPRJ]dopNON%TfRJUNOW-nWJJS l T'^abRg WJRJS'g(vy l J{bi NuR{JSdFmR~VOTVPRNrQCNySab{^'NO^T'K

f1(x1, . . . , xn) = 0,

fs(x

1, . . . , xn) = 0.eabRST'a

Jx =(

fixj

x

)

M(s, n) ^abfxgh{aNPRVdA&I nl .ipoG]suRJK@VPTrVPRNPQCg%na|NPNGJ`'S'N l Nui TRNui cq NOVi TJ^J`'S'N l Nui NP^ q$] %jr s tu( }CVui T~Sab^'LwvOx

Jx = k{VOF l yp^a

XnbRJ%QC^J\pNOVPR{J

Xd'{id'NuRVd

Li/a l W'TQ"QC^'JLPJJHSafxT'NPQSabfOQCNOS'^JVPRTn k ujyaWXVPU'TRabRcSa^LqQabRS'T'g{z%WJHT|} VPT'VPRNOQCJK9ySa{^NO^'T'K VO{bdf]ab^'J

JRJHSab\|NP^'T'NF : Rn(x1, . . . , xn) Rs(y1, . . . , ys)

F (x1, . . . , xn) = (y1, . . . , ys),

yi = fi(x1, . . . , xn)

X = F1(0)i'di FmHJKRJU'W'T

x0 RnJ`'SN l Ni NO^ l T'NOSNO^''T'ai

dx0FJRJHSab\|NP^'Td

F{IsPRJKRJU'WNdx0F : R

n Rs nWJbRJS'ghK{NOWRJS a = (a1, . . . , an) JbRJHS'ab\~aNuR{_{NPWRJSb = (b1, . . . , bs)

`JJSQyiN

bi = (dx0F (a))i =

n

j=1

fixj

x0

ai = (Jx0a)i

:RJ'n]JUNO{T l ^'J'nPi T^'NOK'^JNJRJHSab\|NP^'T'NjxmR{NPS\ l abNPRVdnURJ)vOxJx0 = 3~3 x0F sPRJ{g6RNOWaNuRTfGVOJJR{NuRVuR{yFmoTvqJ`'S'N l Nui NP^'T'K j

%S'T'{N l NPQ y l JH^'ghKVO`'JVOJHq{ghUT'Vui NP^'Td l T'NOSNO^''T'ai/a suRJR[`'ST'NOQUaVPRJ~T'VO`'Ji cfOyNPRVdeT{ l SyLPTvf]a l a]Uav'ujnRJHg {ghU'T'Vi TRc

dx0F (a)nSaVr

VOQCJRS'T'Q9W'S'T'{byF(t)

{R

nn `'SJ]vJ l doyFU'NOS'Nuf

x0TT'QCNOFtoyF{

x0WaVxar

RNuic^'ghKe{NPWRJSa ^aST'VPyKRNS'T'Vuy^'JW-ujJ|NPVPRc

(0) = x0,d

dt

t=0

(t) = a

JL l aRNuv^'T'UNOVOWabd`'S'J{NOS'Wa

dx0F (a) =d

dt

t=0

F ((t))

\ bA](`CJWabfxabRc'nURJLPSy`'`aO(n,R)

d{bid'NuRVd_Li/a l WT'Q@QC^'JLPJJHS'arfxT'NPQ"T^aKRTNONSafxQCNOS^'JVPRc'j

c3'd~P `=Sy`'`aO(n,R)

{bi J]\|NP^a{_`'S'JVuRSab^'VPR{JRn

2 j(JJR^JwpNP^'T'NXXT = 1

l abNPRn2

yS'a{^'NP^'T'K-jC%RJHg l JWaf]abRc'nURJLOSy``aJSRJLOJ^'ai c^'g(vQabRS'Td'{bi'd'NPRVdMQC^'JLPJJHS'abfxT'NPQ `'J l VPU'TRaNPQS'a^'LQaRS'T'gzWJHTsPRJKVOT'VuRNOQCg yS'a{^'NP^'T'K-jGid[suRJLPJ'n=VOJLi aVO^'JJ`T'Vxa^^'JK[{g6wpN~WJ^'VuRSyW'T'T-nJ`'S'N l Nui TQJbRJHS'ab\|NO^T'N

F : Rn2 Rn2 `'JJS'Qy]i N F (X) = XXT 1 jzVP^'J'n'URJ

O(n,R) = F1(0)jb1(ghU'T'Vi T'Q l T'GNPS'NO^'Tai

dX0F : Rn2 Rn2{RJU'WN

X0 O(n,R)ji'dqsPRJLOJSaVPVOQCJRS'T'Q W'S'T'{byF

(t){

Rn2 nRaWyFn

URJ(0) = X0

T ddt

t=0(t) = X

r3^'NOWJRJSabdQabRST'ajnJL l a|T'QCNONPQ

A := dX0F (X) =d

dt

t=0

F ((t)) =d

dt

t=0

[(t)(t)T 1] = XXT0 +X0XT

k UNO{T l ^'J'n URJAr3VOT'QCQCNuRS'TU'NOVPWadqQabRST'aj/:RJJfO^a]UaNPRn URJ 3 X0F`'S'T'^a l i NP\_TR9QC^'J]\|NPVPR{yVOTQCQCNPRST'U'NOVPW'TvQabRS'T'-j[ l SyLOJK VPRJS'J^g%n

l idiFIHJKVOT'QCQCNuRS'T'UNOVOWJK QabRS'T''gAn3QCJ]\^'Jz^aKRTXQabRST'y

XnRar

WyFnURJA = XXT0 + X0X

TnCL l N

X0 O(n,R)jGidsPRJLOJ l JVPRaRJU'^J

`'Jbi J]\_TRcX = (1/2)AX0

jaW'T'QJHSabfxJQhnJHSabf 3 X0F JRJHSab\|NP^'Td dX0F VOJ{`'a l aNuR[V`'S'JbrVPRS'a^'VPR{JQzVOT'QCQCNuRS'T'UNOVOWTv_QabRS'T-j3i N l J{abRNui c^'J'nSab^'L6QabRS'T''gzWJbr

HTVOT'VuRNOQCg%n J`T'VOgh{abFmopNPKLPSy`'`yO(n,R)

n`JVPRJd^'NO^n'RjNO(n,R)

rCLi/a l rWJNGQC^JLOJJHSabfOT'Nj

1(ghUT'Vui TQ"SabfOQCNOS'^'JVPRc_sPRJLOJQC^'JLPJJHSafxTdj%QCNONPQvOx

JX0 = 3~3 X0F = 3~ { VOTQCQCNPRST'U'NOVPW'T'NpQaRS'T'g } =(1/2)n(n+ 1)

j%JsPRJQy 3~ O(n,R) = n2 (1/2)n(n+ 1) = (1/2)n(n 1) j

1RaWJQM\|N l yvNQCJ]\^'J l JWabfxabRc'nURJ{VON`S'T'{N l NO^'^ghN%{g6wpNmLOSy`'`'gd'{bidFmRVdQC^'JLOJJHSabfxTd'QCT-j+

\ bA]` k `'TVxabRcWaVxaRNuic^'JNG`'S'JVuRSab^'VPR{JWSL(n,R)

{N l T'^'T''Nbjc3'd~P `tyVuRc

X(t)Li/a l Wad W'ST'{abdnhiNP\~abopabd {LPSy`'`'N

SL(n,R)T `S'JxvJ l dopabd U'NOSNPfzN l T'^T'y/X(0) = 1

jwJL l a@WaVxabRNui c^'ghK {NPWRJS WsPRJK@W'S'T'{JK@{N l T'^'T'N`'ST'^a l iNP\_TRqWaVxaRNuic^'JQy1`'S'JVPRS'a^'VPR{y/

A :=ddt

t=0X(t) TeSL(n,R)

jCaWpWaW|W'ST'{abdiNP\_TR%{LPSy`'`'Nhy^'T'QCJ l y]idS'^'g(vQabRS'T-nRJ 34.5 X(t) = 1 j:T'NOSNO^''TSydsPRJ_Sab{NO^'VuR{J_`'JbiyU'TQh

0 =d

dt

t=0

34'5 X(t) = v ddt

t=0

X(t) = v AaW'TQ JHSabfOJQ^abwNeWaVxabRNui c^'JN[`'S'JVuRSab^'VPR{JiNP\_TRM{M`'S'JVPRS'a^'VPR{NQabRS'TqV^y]i NP{ghQXVui N l JQhjJWab\|NOQhn/URJJ^'J_V^'T'QVOJ{`'a l aNuRjGidsPRJrLOJ@^'NPJHxvJ l T'QCJ l id"`'SJTfx{Jbi c^'JK"QabRST''g

ARaWJK-n:URJ v A = 0 `JbrVPRSJTRc|WS'T'{yF

X(t){GLOSy`'`'N

SL(n,R)n`S'JxvJ l doyF U'NOSNPfN l T'^'Ty~RaW-n

URJHgA

Hg(iWaVOabRNi c^'ghQ{NOWRJS'JQWsPRJKMW'S'T'{JK@{N l T'^'T'Nj1Wa]U'NrVPR{N~RabWJK1WS'T'{JK[{JbfxcQCNPQ

X(t) = exp (tA)jnNOK'VuR{TRNui c^'J'n

X(0) = 1T

ddt

t=0X(t) = A

j k VuRaNPRVd_`JWabf]abRc'nURJ%W'S'T'{abdi Nu\TR%{LOSy``'Nj1(JVO`'JbicbrfOyNPQCVdeTfO{NOVuR^'ghQXTfiT'^'NOK^'JKai LONPHS'g S'a{NO^VPR{JQ

3b4.5 exp (tA) = exp v (tA)k RVOF l aVui N l yNuRnURJ 3b4.5 X(t) = 1 jRaW-nOWabVxabRNui c^JN:`S'JVPRSa^'VuR{JtWLOSy`'`N3y^'T'QCJ l y]idS'^'g(vGQabRS'T'VOJ{br`a l aNuR%V(`S'JVPRSa^'VuR{JQzQabRS'T'|V:^y]i NO{ghQ[Vi N l JQhj

\ bA]`JpJWaf]abRc'n:URJ@LPSy`'`aSU(2,R)

LOJQCNOJQCJSG^'azRS'NvQCNOS'^'JKVONOS'N

S3j

c3'd~P `=m^ai Tf_yVui J{T'K-nZWJRJSghQ l Jbi\_^ay l J{biNPR{JSdRcQabRST'aTfSU(2,R)

`'JWafxgh{aNuRnURJ`'S'JTfx{Ji c^'ghKpsi NOQCNP^R%LPSy`'`'gMf]a`T'VOgh{abNPRVd{I{T l N ( z w

w z

) nL l Nz, w C RaW'TNnURJ |z|2+|w|2 = 1 j*JtNOVuRc (z, w)

S3 C2 j

?CP'PMbP-bPpxxCOPuZ'PbPu=pObP*xxx Oxbbxu'uuuP]u9uuPPu-' PbuuPu P'ZPmuOuP=[u= uub[P~uPub'Iueb' 'Z~xx 'PPuub]APuP('uPuP(u=OPECT.*AI[I

#/$%bzB' 2\ bA]a`CJWabfxabRc'nURJLPSy`'`a

U(n,R)d'{id'NuRVd_Li/a l W'TQ@QC^'JLPJJHSar

fxT'NPQ"T^aKRTNONSafxQCNOS^'JVPRc'j\ bA]C`%VPVui N l J{abRc

GL(n,R), SL(n,R), O(n,R), SO(n,R), U(n,R), SU(n,R)^aWJQC`'aWR^JVPRcTVO{dfx^'JVuRc'j\ bA]`JWabfxabRc'nURJLPSy`'`a

SO(3,R)LOJQCNPJQCJS'^a_RS'NuvQCNPS'^'JQy

`'S'JNPWRT'{^'JQy`'S'JVuRSab^'VPR{yRP 3

j 1ZC5 0 -B'

9 > :2 jmNPwNO^'TNm`'Ji ^'JVuRcF9VOJ{`a l abNPRGVhS'NPwNO^'TNOQ-a l a]U'T nRJbicWJ{^'NPWJr

RJS'g(vqQCNOVPRav[^a l J`JVPRa{TRcU'NOSRyVOJ`'Sd\|NO^'TN]uj1 suRJQ Vui'yU'aN|JHSafl T'NOS'NP^''Tai aJbRJHS'ab\|NO^Td

F (X) = XXT 1 `'J l VOUTRa^'^'JLOJ{_i FmHJKRJU'WNqy^'TRaS^'JK LOSy`'`'g Hy l NPRVPJ{`a l abRcVe`S'JVPRSa^'VuR{JQ sOSQCTRJ{g(vQabRS'T

XT = XujC%JsuRJQyLPSy`'`a1y^'TRaS'^'g(vQabRST'Hy l NuRLi a l W'T'Q

QC^'JLOJJHSabfOT'NOQSabfOQCNOS'^JVPRT 3~ U(n) = 2n2 3~ { sOSQCTRJ{g QabRST''g } =n2j jxJ l SJH^'JNCJ`'T'Vxab^'T'N3Wi/aVPVOT'U'NPVOW'TvQaRS'T'U^'g(vLOSy`'`GQCJ]\^'J^abKRTG{

W'^'T\_WN:j:jpJQCNP^'WJ "[ua&|n].P%$O.&I ($/ n*i/ab{a%n j%T\pN`'S'T'{N l NO^'aRaHbiT'a~JR{NuRJ{=Sy`'`'aEGT y%JQC`abWR^'JVPRc ({bdfx^JVPRc mabfOQCNOS'^'JVPRcGL(n,R)

r rn2

SL(n,R)r

n2 1O(n,R)

rn(n 1)/2

SO(n,R)

n(n 1)/2U(n,R)

n2

SU(n,R)

n2 1

0j3:Ra1f]a l a]Ua1SabfOHT'Sai aVOcz^a[i NPW''Tdvj(}opNNONS'NPwNO^'TNeQCJ]\^'J1`'JbrVOQCJRS'NPRc{W'^T\WN|j:jpJQCNP^'WJ'nj,mjyWV- hO%&~%n].P%i&~% / n nVPRS-jZ j

) 2 *+ q# &'# A :z

X

@C

> 2 #(1Z < # ! 0 1ZC2

k `'S'N l NuiT'QLi/a l W'T'NJRJHSab\pNO^'Td~Li a l W'Tv|QC^'JLOJJHSabfOT'K-n`S'N l VuRa{bidrFmopTNNOVuRNOVuR{NP^'^'JNJHJHopNP^'T'N l T'GNPS'NO^'T'SyNOQCg(v%y^'W''T'KM{a^aiTfxNbj%yVPRc

MnnNm

r l {aLi/a l W'Tv QC^'JLPJJHS'abfxTdj)maVPVOQabRS'T'{adMn

TNmWabW_RJ`'Jbi JLPT'U'NPVOW'T'N%`S'JVPRSa^'VuR{anQCJ]\^JLPJ{JS'TRcJG^'NP`'S'NOSgh{^'g(vJbRJHbr

Sab\|NP^'Tdvf : Mn Nm ji/a l W'T'NVuRSyWRbyS'g%nCf]a l ab^'^'ghN^aQC^'JLOJJHSarfxTdvn`'Jbfx{Jbi'd'FmRz{{NPVPRTHJbiNONyfOW'T'KWi/aVPVJRJHSa\|NO^'TK-j k RJHSa\|NO^'TN

f : Mn Nm NOVPRNOVuR{NP^'^'J_fxa l abRc~{i JWai c^g(vWJJS l T'^'abRavj %QCNO^'^'J'NOVri Tx Mn r6`S'JTfO{Jbi c^'abdRJU'Wan (U, ), (V, ) r6WaSRg {QC^'JLPJJHSafxTdv

Mn, NmVPJJR{NPRVuR{NP^'^'JRaW'T'NbnURJ

x U n f(x) V nbT W (x) r-JW'S'NPVPR^'JVPRcRJU'WTxRabWad-n URJ

W (x) U n f(W (x)) V WJRJSabdVPyoNOVPR{yNuR{_VPTiy^'NO`'SNOS'gh{^JVPRTf{xjnJL l a~JbRJHS'ab\|NO^T'N

f1 : (W (x)) (V )^abfxgh{aNPRVdqA&|/uA'&|#Pu &~]* u($

f{JW'S'NPVPR^'Jbr

VPRTRJU'W'TxjbabWJN%`'S'N l VPRa{biNO^'T'NG`'JfO{JbidNPR~`'ST'{bi NPU'c~`'J^dRT'NLi/a l WJLOJ

JRJHSab\|NP^'TdRn

{Rm

nTfOyUaNPQCJNG{_ab^ai TfONj 3' Z%k RJHSab\|NP^'T'N

f : Mn Nm ^abfOgh{aNuRVdLi/a l W'T'QJRJHSab\|NP^'T'NOQ{JW'S'NPVPR^'JVPRTMRJU'WTx Mn nNOVi T@^'NPWJRJSJN[aRJL l aTi FIHJN{~VOTiyLi/a l WJVuRTq%y^'W''TKVOWi NOK'WT WJJS l T'^abR^'JN`S'N l VuRa{bi NP^'T'N

f{%JW'S'NOVuR^'JVPRT

xd'{bidNPRVd_Li a l W'T'QzJRJHSab\|NP^'T'NPQzJHi/aVPRNOKNO{Wi T l J{g(v

`'S'JVuRSa^VPR{jk R|i JWai c^'g(vJ`'S'N l NuiNO^'T'K`'NOSNOK l NPQ"WLi JHaic^'ghQhj 3' Zk RJHSab\pNO^'T'N

f : Mn Nm QC^JLOJJHSabfOT'K^abfOgh{aNPRrVd %nNOVuiTJ^'J~Li/a l WJN{~JWS'NOVuR^'JVuRTqWab\ l JKRJU'W'Tx Mn j19RNOJS'T'TqLi/a l W'TvQC^'JLOJJHSabfOT'KLi a l W'T'NJRJHSab\|NP^'TdT'LOSabFmRRby\pN

S'Jbi c'nURJWaWyFT'LPSaFtR^NO`'S'NPS'gh{^'ghN~JRJHSa\|NO^'Tdq{JHoNOKqRNPJS'T'TqRJr`'Jbi JLPT'U'NPVOW'Tv`'SJVPRSa^'VuR{'j

%S'JVuRNPKwpT'Q[`'ST'QCNOS'JQqLi/a l WJLPJ%JRJHSab\|NP^'TdQCJ\pNPRViy\TRcIRJ\ l NrVPR{NO^'^'JN%JRJHSab\|NP^'T'NILi a l WJLOJ|QC^JLOJJHSabfOTd^'aVONPHdjby%JQC`JfxT'Td l {byvLi/a l W'TvJRJHSab\|NP^'T'KRaW\|Nd'{bi'd'NPRVdLi a l W'T'QJRJHSab\|NP^'T'NPQhj D %x%'uuuP>I

RNPbbxP%uuuub:]Px'h(h(bO]mu'bAKOMn

Nm

/'PPuubtRN

RM

uuPuubumPuP(f : Mn Nm

=PuuGbbuuuub:PO%uux Mn

x3Px6u=u:':PU RN

x U

bxuPuP(f : U RM

buP:f

U Mn

0

\ bA]^a`yVuRcT2

RJS|{R3

nb`'JiyU'NO^'^ghK{SabopNP^'T'NPQzJWSy\^'JVPRT(x

2)2 + y2 = 1{JWSyLGJVOT

yj' `yVPRc

(a, b, c)WJJS l T'^aRg {^NPwp^NOK^JS'Qai T

WT

2 {RJUWN p jfJL l an `'JVOWJbi cWy a2 + b2 + c2 = 1 n RSJK'Wy (a, b, c) QCJ\_^'JSaVOVPQabRST'{abRc_WaWWJJS l T^abRg9RJU'W'T^a l {yQCNPS'^'JKVONPS'Njn1:JfxcQCNPQsPRyRJU'Wyf]a

f(p)j(JWa\TRNnURJ

fNPVPRcLi/a l WJNJRJHSa\|NO^'TNRJSa~{~VONOSyj

c3'd~P `LJWab\|NOQhnZURJqJRJHSab\|NP^'T'Nfd'{id'NuRVdMLi/a l W'T'Q {eJWS'NOVuRbr

^'JVPRTqRJU'WTA = (3, 0, 0)

jGid l SyLPTveRJU'NPW l JWabf]abRNui cVPR{Jab^ai JLPT'U'^'JvJbR]dXTfJ`'S'N l Nui NP^'TdXJbRJHS'ab\|NO^TddVO^'Jn3URJ@NPVui T"J^'JzLi a l WJ@{J l ^JKRJU'WNbn RJLi/a l WJ_T{VOF l y^a|RJS'NbnRjWyVuRS'JNP^'J{Nuf l NJ l T'^'aWJ{Jj

1(JbfxcQCNPQ RJU'Wy@^aJW'Sy\^JVPRT(x 2)2 + y2 = 1 {e`iJVOWJVuRT (x, y) nl id1WJRJSJK1iyU-n=^'a`'Sa{i NO^'^ghKW1^'NOK1Tf'NO^RSaJW'Sy\_^'JVPRT-nZJHSafOyNuR

yLPJbiVI`'Jbi J]\_TRNi c^'ghQ^a`Sa{bi NP^'T'NPQJVOT

xjSJ{N l NOQU'NOSNPf%suRyRJU'Wy

JW'Sy\_^'JVPRc'n`aSaii Nui c^yF `i JVOWJVPRT(x, z)

T~i NP\~aoyF ^aRJSNnaGf]aRNOQ{JfxcQCNOQRJU'Wyn:VOQCNuopNO^^yF ^a1yLOJi

`'JsPRJK"JW'Sy\_^'JVuRT ^'aS'T'VuyKRN

S'T'VPy^'JW- jLJL l a`Jbi J]\|NO^T'NpRJUW'Tx T2 QCJ]\^'Jfxa l a{abRc`aSJK (, ) j-aQCNPRT'QhnURJsuRJG^'NhNOVuRcGVOT'VuRNOQaGWJJS l T'^aR^'a{VONOQ[RJS'NnRj W l id l a^'^JK

RJU'WTxyLi g

TJ`'SN l Ni NO^'gRJbi cWJeV~RJU^'JVPRcF l J'NuiJLOJeWSabR^Jbr

LOJ2

jNPQ^NQCNO^'NPN(, )

QCJ]\_^'Je`'S'T^dRc1fxai JWai c^ghNWJJS l T'^abRg {JW'S'NPVPR^JVPRTeRJU'WT

A (0, 0) jEGNOLPWJ_`'J l VOUTRabRcnURJ{~JHYZNPQi FmoTvWJbrJS l T'^abRav(x, y, z) = r R3 {|JW'S'NOVuR^'JVPRTRJUW'T A RJSHy l NuR~f]a l ab{abRcVdVui N l yFmopTQXJHSabfOJQ

x = (2 + cos ) cosy = sin z = (2 + cos ) sin

k RVOF l aQCJ]\^'J~{ghUT'Vui TRcWJJS l T'^abRg{^'NPw^'NOK^'JS'Qai T-n = r r

j

n = (cos cos, sin , cos sin) S2k RJHSa\|NO^'TN

f`'NPS'NO{J l TRzRJU'Wy

A{qRJU'Wy

(1, 0, 0)VONPS'g%jJ1 Wa]U'NPVPR{N

WabSRg ^a1VONOSNnVOJ l NPS\|abopNPKRJUWy nnunCQCJ]\_^'J[{fOdRc(S21+ = {x >

0}, (x, y, z) = (y, z)) j:JL l aGWJJS l T'^abR^'JN%`'SN l VuRa{bi NP^'T'NJRJHSab\|NP^'Td f{_JWS'NOVuR^'JVuRTRJU'W'TAHy l NPR_T'QCNuRc{T l

f (, )1 : (, ) 7 (sin , cos sin),WJbRJS'JNnCJU'NP{T l ^J'nd'{bi'd'NPRVdLi/a l W'TQJRJHSab\pNO^'T'NPQ JHbi/aVuRNOKNO{Wi T l Jr{g(v`S'JVPRSa^'VuR{'j'JsPRJQyJRJHSa\|NO^'TN

fd{bidNPRVdLi/a l W'TQ{JW'S'NPVPR^'Jbr

VPRTRJU'W'TAj%Jp`S'T'U'T'^'aQhnJHVuy\ l a{bwpT'QCVd{|VOaQCJQ^a]UaiN l JWabf]aRNuicbr

VPR{an J^'J|d'{bi'd'NPRVdLi a l W'T'QT^'a~{VONOQRJSNj

1 QabRNOQabRT'WN[{VONM]*Pa&| S'aVOVPQabRS'T{aFmRVd VRJU'^JVPRcF l JM^'NPWJrRJS'g(v u`'S'NOJHSabfOJ{a^'TKZ(jCOG#jG:RT`S'NOJHSabfxJ{a^'Td VOJxvSa^dFmRVO{JKVPR{aeJHYZNOWRJ{'nZWJRJS'ghNQCgTfPyUaNOQhjaWnQC^J\pNOVPR{aeSaVPVOQabRS'T'{arFmRVdeVRJU'^'JVuRcFHT'NOW'T'K-nLPSy`'`'g VRJU'^'JVPRcF l JLPJQCJQCJS'TfOQCJ{'nRJr`'Jbi JLPT'U'NPVOW'T'N|`S'JVPRSa^'VuR{aVRJU'^'JVPRcF l JLOJQCNOJQCJSGTfxQCJ{TR l jLmJi cRaW'TvP`S'NOJHSabfxJ{a^'T'K=P{%VuiyUabNtLi a l W'Tv|QC^'JLOJJHSabfOT'K~T'LOS'aFmR l T'NOJbrQCJS'TfxQCg%j

k U'NP{T l ^'JnLOJQCNPJQCJS'TfOQCg ^'NVOJxvSa^d'FmR_VPRSyWRySyLi/a l WJLOJQC^JLOJbrJHSabfOTdvJR]dWaSRa`'Jbr`S'NP\_^'NOQy`'NOSNuvJ l TRp{WaSRynLi/a l WJVuRcy^'W'T'KVOWi NPK'W'T"S'abfxSywpaNPRVd uj(%JsPRJQyMRS'NPHyNuRVd"yVOTiTRc@J`'S'N l Nui NP^'T'NLPJQCNOJrQCJS'TfxQa|RaW-n URJHg Li/a l WabdVPRSyWRySaVPJxvS'a^di aVOc'j

3' Z yVuRcf : M N LOJQCNOJQCJSGTfxQ Li a l W'Tv1QC^JLOJJHbrSabfxTK-j3}Vi TJRJHSab\|NP^'T'Nfd{bidNPRVdLi/a l W'T'QhnT-nHJbi NONRJLOJ'nJHSabR^JN

JRJHSab\|NP^'T'Nf1

RaW\|Nd'{id'NuRVdLi/a l WT'QhnRJ[JRJHSab\pNO^'T'Nf^abfxgh{ar

NPRVd jjOG(jCOj:}CVui TVPyopNPVPR{byNPR l T'NOJQCJS'TfxQM

{NnRJ

LOJ{JSdRn URJQC^'JLPJJHSafxTd[jjO(jCCj+\ bA](`%S'T'{NPVPRT`'S'T'QCNPSLi/a l WJLPJpLOJQCNOJQCJSGTfxQanWJRJS'ghK^'Nd'{br

id'NuRVd l T'NOJQCJSGTfxQCJQhjc3'd~P `fmaVPVOQCJRS'T'Q JRJHSab\|NP^'T'Np`'SdQCJK[VOJVuRa^ l aSR^'JKzVPRSyWRyr

S'JKLi/a l WJLPJQC^'JLPJJHS'abfxTd^'a~VPNOH]d-nfxa l a{abNOQCJN%JS'Qy]i JK

f : R R f(x) = x3

zVP^'J'nmURJ 8 rLi a l W'T'KLPJQCNOJQCJS'TfxQhj kml ^aWJ'nIJHSabR^'JN@JRJHSa\|NO^'TNf1(x) = 3

x^'Nd{bidNPRVdLi/a l WT'Q Rj W"^'NT'QCNONuR@`'S'JTfx{J l ^JK"{1^y]i N]j

JsPRJQyf^Nd{bidNPRVd l T'NOJQCJS'TfOQCJQhj

({JK'VPR{aQC^JLOJJHSabfOT'K-nZVOJxvSa^dFmoT'NOVd`'S'T l T'GNPJQCJS'TfOQav1^arfxgh{aFtRVd1jj3/PnlO%e#u bn&|]%j ktl ^'T'QTf%RabW'TvT^'{aS'Tra^RJ{d{bid'NuRVdSabfOQCNOS'^'JVPRc_QC^'JLPJJHSafxTdj

\ bA]`C-a l a l T'Q^'at`'SdQCJKVJHghU'^'JKRJ`'Jbi JLPT'NOKZVuRSyWRbySypLi/a l rWJLOJQC^'JLPJJHSafxTd l {byQdS'abfx^'ghQCTGVO`JVOJHaQCT|xVC`'JQCJbopcFMWaSR

(R, (x) =x)

T(R, (x) = x3)

VOJJbR{NuRVPR{NO^^'J'j)JWab\_TRNnURJsPRT QC^'JLOJJHSabfxTdl T'NOJQCJSG^g%j PuPxPhbb]'PuPuu>%PuuP.6REUWI*='uu'GOPu'>O(u3CO'u''PuPuu>RbuL>(buP:36O'O'u-uuuu]Puh:bP]ubPuu3uub6b6uOB(3u(u(uu:xP'=:*CmP(uu'ObO6'PPuubGO:uPhOu%ueb

Cn xx

(n 1)tbhx3PuhPOP'u%bb]xxbxub'=hb'|bb'uuPuu]GPu

R>%PPbCP'ubC?C'PuPCbuxxeRn

c3'd~P `JWab\|NPQhnURJpJRJHSa\|NO^'TNf : R R nJ`S'N l Ni NO^^'JNVi N l yrFmopTQJHSabfOJQ

f(x) = 3xd{bidNPRVd l T'NOJQCJSGTfxJQ@yWabf]ab^'^'g(vQC^JLOJbr

JHSabfOT'K-jR

f R R RNP`'S'NPS'gh{^'JVuRc

fJUNO{T l ^ajy%JJS l T^abR^JN`S'N l VuRa{bi NP^'T'N

fT'QCNPNPR_{T l

f1(x) = f(x) = ( 3x) = x

:RJ9Li/a l WJNJRJHSab\|NP^'T'N@NP{Wi T l J{g(v `'S'JVuRSa^VPR{n%`'JsPRJQy 8 rLi/a l rWJNJRJHSa\|NO^'TNQC^'JLOJJHSabfxTK-jy%JJS l T'^aR^'JN`'S'N l VPRa{i NO^'TNpJHSaR^'JLPJJRJHSab\|NP^'Td

f1(x) = x3T'QCNPNPR_{T l

f11(x) = f1( 3x) = (x) = x

JsPRJQyf1

RJ]\|NLi a l WJNGJRJHSab\|NP^'T'NnVi N l J{abRNui c^Jfr l T'NOJQCJSr

GTfxQhj1QabRNOQabRT'U'NOVPWJQ a^ai TfxN'NO^RSaic^'JKT l NONPKd{bid'NuRVd`'NOSNuvJ l JbR

l T'NOS'NP^''T'SyNOQCg(vey^'W'T'KJHopNPLOJ_{T l a|WHJbi NPNyfOWJQyWi/aVPVPyr3QC^'JbrLOJUiNO^aQ^a]Uai c^'ghQXJRSNPfxWaQXVPJJR{NPRVPR{byFtopNPLOJSd l ajf1 `'S'JVuRNOKwpNOQVuiyUaN`'NOSNuvJ l JVPyopNPVPR{id'NuRVd9WVOJJR{NPRVPR{yFmopNPK"i T'^'NPK'^'JKy^'W'T'Tr l TGNPS'NO^'Taiyj/(JJR{NPRVPR{yFmopTK QCNPRJ l ^abfOgh{aNPRVd9 uCl .ij19ViyUaNLi/a l W'TvQC^'JLPJJHS'abfxT'Kn l id`'SJTfx{Jbi c^'JLOJ~Li/a l WJLPJ_JbRJHS'ab\|Nur^'TdRJ]\|NQCJ]\_^'J`'JVPRS'JTRcNOVuRNPVPR{NO^'^'JN|iT'^'NOK^'JNJbRJHS'ab\|NO^T'Nj k ^'J_RJT^afxgh{aNuRVd l T'NOS'NP^''Tai JQhj

yVuRcF : M N Li/a l WJNIJRJHSa\|NO^'TNT x M `'S'JTfx{Jbic^abdRJUWa^a

Mj/JL l aQCJ]\_^'JMJ`'SN l Ni TRcVui N l yFmoNONNPVPRNOVPR{NO^^'JN@JRJHSa\|NO^'TN

WabVxabRNui c^JLOJ_`'S'JVuRSa^VPR{aTxM

{~WaVxaRNuic^'JN`'S'JVuRSa^VPR{JTF (x)N

j

dxF : TxM TF (x)N O' n = 4 ]hbuu*3bCP'P='uuuP]9xPbxx@

Snn = 1, 2, 3, 5, 6, 12

uIbh'uuPuu]Puu]:Pbu'P'uuuP]G' 4 6P'O'uPuG'uuPuu>u6]tbCP'ubP u]PL>quuP'x3PxqO'O'u=PZ|bCu'P=='PuPuu]ALBPPuP P-( D %@%]buOMZ.(Pxux3PuPumPbumhP'Pbuxuuu6uu

x

u u=ss/3' %Jbi cfPyNOQCVdGJ`S'N l Ni NO^T'NOQeWaVOabRNi cbr^'JLOJ[{NPWRJSanWaWJ`'NPSa'TT l TGNPS'NO^'T'S'J{ab^'Td-j}Vi T

v TxMnh

C(N)r3Li/a l Wad%y^'W'Td^'a

Nn'RJ

dxF (v)(h) = v(h F ) ss3uL=sks/3' yVPRc(x1, . . . , xn)

T(y1, . . . , ym)

i JbrWai c^'ghNWJJS l T'^'abRg {qJW'S'NOVuR^'JVPR]dvRJU'NOW

xTf(x)

VPJJR{NPRVPR{NO^'^'Jj:`yVuRc

yi = yi(x1, . . . , xn)ni = 1, . . . , m

rZWJJS l T^abR^JN`'S'N l VPRa{biNO^'T'NmJRJHbrSab\|NP^'Td

FjbJL l a

(dxF (v))i = y

i

xjvjj

st%n= Lss 3' Z yVPRc(t)

rGLi/a l WabdW'S'T{abd^aM

RaWabdnURJ(0) = x

T(0) = v TxM

jJL l adxF (v) =

ddt

t=0F ((t))1UabVPR^JQViyUaNLi/a l WJK%y^'W''TT

f : Mn R ^aqQC^'JLPJJHS'abfxT'TnQCg {Wab\ l JKRJU'WNx M T'QCNONPQ i T'^'NPK'^'ghK%y^W''T'J^'ai dxf : TxM =

Rn Tf(x)R = RjeabRS'TasuRJLPJ%y^'W''TJ^ai/a{i JWai c^'g(v1WJJS l T'^ar

Rav(x1, . . . , xn)

^'aMn

r:sPRJVPRS'JWan/VOJVuRJ]dopabdeTfUaVuR^'g(v`'S'JTfx{J l ^g(vf

xi

j:RJRp%y^'W''TJ^ai^abfOgh{aFmR un&~ %y^'W'T'Tf{%RJUWN

xT~JHJr

fx^a]UaFtRUNOS'Nufv 2 3 f |x j\ bA]`maVOVPQCJRST'Q9JRJHSab\|NP^'T'Nexp : M(n) M(n) nL l N M(n) =

Rn2

r3`S'JVPRSa^'VuR{JW'{a l S'abR^'g(vQaRS'T'j

expX = 1 +X +X2

2!+X3

3!+ . . .

k `T'VxabRc l T'GNPS'NO^'Tai@JRJHSab\|NP^'Tdexp

{^y]i Nbn-RjNd0 exp : T0M(n)

T1M(n)j

c3'd~P `3y l T'Q `Jbi cfOJ{abRcVdLONOJQCNPRST'U'NOVPW'T'Q J`S'N l Ni NO^T'NOQ l T'rGNPS'NO^'Tai/ajmaVOVPQCJRS'TQ {

M(n)W'S'T{yF

(t)RabWyFntURJ

(0) = 0T

u3Pb:b'xbxu=OPZPuP3ubP]uZ- 'u6P'uuh]h|B COQ.S*I*I*/NOI*P(PQRG*SAIARC.G*ER3CI*IOP':buux=R>L>(x=uPuu]umP'O hx(hOO_ubRKbxbux3OPu-tu'>b6]u=uu:uPu:]uBu6Ox%ObuPmu=X

(0) = XjnJL l a

(d0 exp)X =d

dt

t=0

exp((t)) = X

aW'TQ JHS'abfxJQhn l T'NOS'NP^''Tai@sOWVO`J^'NO^'Tai c^'JLOJJRJHSab\|NP^'Td1{^y]i N`'S'N l VPRa{bi'd'NPRVOJHJKRJ]\ l NPVPR{NO^'^JNJRJHSa\|NO^'TNj

k VOJHghKT^RNPS'NOV`'S'N l VPRa{bi'd'FmR^'NOWJRJS'ghNVP`'NOTai c^ghN{T l g JRJHSar\|NO^'TK-j

3' Z i/a l WJNJRJHSa\|NO^'TNf : M N ^'abfxgh{abNPRVdp ha u %nCNOVi T l T'GNPS'NO^'Tai

dxf : TxM Tf(x)Nd'{id'NuRVdQCJ^'JQCJSr

GTfxQCJQ l idi FIHJKRJU'WTx M x M vOx (dxf) = 3~ M uj 3' Z %JLOSy\pNO^'T'N

f^'abfxgh{aNPRVd#O/Z u NOVi T@J^'Jd'{r

id'NuRVd LOJQCNPJQCJS'TfOQCJQ ^'aVP{JK JHSafj k HS'abf[{bi J]\|NP^'Td9^abfOgh{aNuRVd P% u] %j

Vi J{T'NQCJ^JQCJS'^'JVuRT l T'NOS'NP^''Tai/azJRJHSa\|NO^'Tdf : M Na{RJQabRT'UNOVOWT{biNOU'NuR~Vui N l yFmopNPNVOJJR^'JwNO^'T'N%^'apSabfxQCNPS'^'JVuRTQC^'JLPJJHbr

SabfxTK- 3~ M 3b~ N j1(VPNm`JLOSy\|NP^'Tdi JWai c^JyVuRS'JNO^'g WaWVuRa^ l aSR^'JNI{bi J]\|NO^T'NRn

Rmn'`'NPS'NO{J l dopNPNRJU'Wy

(x1, . . . , xn){|RJU'Wy

(x1, . . . , xm, 0, . . . , 0)j

k HSafQC^JLOJJHSabfOTd`S'T`'JLOSy\|NP^'T'Te^'NJH]dfxa^Hg6RcQC^'JLOJJHSabfxTNOQ VOQhj-a l a]Uy j k HSabf

f(M)QC^'JLOJJHSabfxTd[`'S'T[{bi J]\|NP^'T'T-n-Rj N~`'J l QC^'JLPJbr

JHSabfOT'Nn'd{bidNPRVdLi/a l W'T'Q"QC^'JLOJJHSabfxTNOQhn l T'NOJQCJS'^'ghQMj

\ bA]a`mabVOVOQCJbRS'T'QVPNOQCNOKVPR{JMJRJHSab\|NP^'T'K`SdQCJK"{z`i JVPWJVPRcnf]a l ab^'^'JNRaW

fa : t 7 (t2, t3 + at)j %S'TWaW'Tvfx^'aUNO^'Tdv`aSaQCNuRSaJRJHbr

Sab\|NP^'T'NHy l NPR_`'JLPSy\|NO^T'NOQhn/a~`'S'TWabW'Tv{biJ\pNO^'T'NPQ|c3'd~P `eabRS'Ta l TGNPS'NO^'Tai/aJRJHSab\|NP^'Td

faT'QCNONuR{T l

dtfa =

(

2t3t2 + a

)

k RJHSa\|NO^'TNfa

Hby l NPR`JLOSy\|NP^'T'NPQ[RJL l aITGRJbicWJmRJL l anWJL l avOxdtfa =

1l id{VONv

t R j3i N l J{abRNi c^'Jn fa `'JLOSy\pNO^'T'N|`S'Tq{VONv a 6= 0 jnpai NONbnWabW~i NPLOWJ`S'J{NPS'TRcnJbRJHS'ab\|NO^T'Nfa

Hy l NuR{i J]\|NO^'TNOQzRJbi cWJ`'S'Ta > 0

j}Vi T

a < 0nRJW'ST'{abd

fa(R)T'QCNPNPRVxaQCJ`'NOS'NPVONPU'NO^'TNq `JVPRS'JKRN_LPSaT'W

sPRJK|W'S'T'{JK`'S'Ta = 1, 0, 1 l idpHJbi cwNOKp^abLid l ^JVPRTpRJLPJ'nbWaWyVPRS'JNP^'JsPRJ_JRJHSab\|NP^'T'Nxuj

FOu6P=P6ZO3ubb:>*KOf : M N

Pb':uPPuuOuP(ux|Puuf(M)

P'x3N%u

fu:P

#/$%bzB' 2\ bA]C`GabKRTJHSaf_PWaVxabRNui c^'JLOJ{NPWRJS'a

v =

(

0 10 0

) `'J ll NOKrVPR{T'NOQ l T'GNPS'NO^'Tai/asOWVO`J^'NO^Rg {|RJU'WN

X =

(

00

)

dX exp : TXM(2) Texp XM(2)

\ bA]`%yVPRcM

r^'NPWJQC`aWR^'JNQC^'JLOJJHSabfxTNnf : M Rn {f]arT'QC^'JJ l ^Jfx^a]U'^JN_V~JHSabfxJQ `'JLPSy\|NO^T'Njz%{bidNPRVd[i TzJ^'J{bi J]\|NO^T'NOQ|

VOS'a{^'TRNJbR{NuRVPJ_VP^'JVOWJK1]\ bA]`(z%{bi'd'NPRVd|i T_JbRJHS'ab\|NO^T'N

exp`'JLOSy\pNO^'T'NPQ|m9{bi J]\|NP^'T'NOQ|

x

*+ 2 */.z? # 7 f=#/&':5 < 7 "X7 &'# 7 &' > [# $ > 2

NOJQCNuRS'Td9`J{NOSv^'JVuRNPK {MNP{Wi T l J{JQ`'S'JVPRS'a^'VPR{NnWJRJSabdXHg(i/aTfx{NPVPR^aNuopNa]yVOVuynd{bid'NuRVd`'SN l QCNuRJQTfOyU'NP^'Td[ un].P%iejj./PnlO %i_uP.&I jasPRJQe`'S'JVuRJQqTG^aLi'd l ^'JQeQabRNPS'TaiN(QCJ]\^J`'J^dRcVPyRc l T'NOS'NP^''Taic^'JLPNOJQCNuRS'TU'NOVPWJLOJ`'J l vJ l aj

k VO^J{^'ghQJHYZNPWRJQTfOyU'NO^Td l id^abV%Hy l yR l {yQCNOS'^ghNG`J{NOSv^'JVuRT{_VuRa^ l aSR^'JQ`'S'JVuRSa^VPR{N

R3j

a`'JQC^'T'Qhn'URJ{VPNW'S'T'{ghNGVQCNPRS'T'U'NPVOWJKRJU'W'TfxS'NP^'TdJ l T'^aWJ{g {Vui N l yFmopNPQVOQCghVi N^aqWab\ l JKS'NOLuy]id'S'^JKWS'T'{JK W'S'T'{adS'NOLuyi'd'S'^anNOVuiTq{NOWRJSqVOWJSJVPRT1^T'L l Np^'NJHSabopaNPRVdq{^yictQCJ]\_^'J{{NPVPRTeRaWyF`aSaQCNuRS'Tf]a'TF^abRbyS'ai c^yFun=URJ l i T'^aJRS'NPfOWaW'ST'{JK[QCNP\ l yRJUrWabQCT-nVPJJR{NPRVPR{byFtopT'QCTfO^a]U'NO^'Td'QM`aS'aQCNPRSa

t1Tt2n{ghU'T'Vid'NuRVdJ l Tr

^aWJ{J'J^aSa{^a |t1 t2| j:1RNPJS'T'T`J{NOSv^'JVuRNPK`'J l JH^'g(vS'NPfPyicRaRJ{^'NPRjma`'S'T'QCNPS-n^aVPRa^ l aSR^JKVONOSNz^'Ni cfPd {{NOVPRT WJJS l T'^'abRgDRaWnURJHg JS'Qy]i g l id{ghU'T'Vi NO^Td l i T'^W'S'T'{g(v^aqVONOS'NVuRai TRabW'T'QCT\|N|WaWq^a`i JVPWJVPRT-jLaWT'Q9JHSabfOJQhn`J{NOSv^'JVuRT1JHbi/a l aFmR^NOWJRJS'g(rQCT#bnha&I P%#b n&|xnRj NT'^'{aSTa^RaQCTn3^Nfxa{T'VdopT'QCT"JbRWJJS l T'^abRj%J{NOSv^'JVuRTrsPRJp`'S'JVuRNPKwpT'N%JHYZNOWRg%n^'aWJRJS'g(v{Jfx^'TWarNPR_{^yRS'NP^'^ddLPNOJQCNuRS'Tdj

%J]\~aiyK-n(VxaQCghQ WJS'JRWT'Qhn^'J`S'T suRJQ WJoy^'VuR{NP^'^'ghQ d'{bi'd'NPRVdVui N l yFmopNPNpJ`'SN l Ni NO^'TNj

3' Z cO@hb$ %i #b: .&a{R3

^abfOgh{aNuRVd l {yQCNOSr^'JNLi/a l WJNG`'J l QC^'JLOJJHSabfxTN

R3j

`'JQCJopcF9RNPJS'NPQCg J^'NPd{^'g(vy^'W'Tdv_yRJU^'T'QsPRJpJ`'S'N l NuiNO^'T'Nbj 3' Z e^'J]\|NOVuR{J

R3 ^afxgh{aNuRVdS'NPLPy]idS'^'JK`'J{NOSvr^'JVPRcFn NPVui T{gh`'Jbi^d'NuRVdeJ l ^J_TfGVi N l yFtopTvyVi J{TK-j*1 l JVPRabRJU'^'J%Qai JKpJWS'NOVuR^'JVuRT~Wab\ l JKVP{JNPKRJUW'T

{g l Nid'NuRVd

WabWXQC^'J]\|NOVuR{JM^y]i NPK"RaWJK9Li/a l WJK9%y^'W''TTF (x1, x2, x3)

n:URJM{@sPRJKJW'S'NPVPR^JVPRT F

x36= 0 `'JVui N`'J l vJ l dopNPKe`'NPS'NO^yQCNPSa'T'TWJJS l T'^abRj j1 l JVuRabRJU'^'JQai JKXJW'S'NOVuR^'JVPRT Wab\ l JKXVO{JNOK"RJU'W'T

f]a l aNPRr

VdWabWLPSaT'WMLi/a l WJLOJ1JRJHSab\|NP^'Tdx3 = f(x1, x2)

`JVui N`J l vJ l doNOK`'NOS'NP^yQCNOSab'T'TeWJJS l T^abRj

j1 l JVuRabRJU^'JqQai JKzJW'S'NPVPR^'JVPRTWa\ l JKzVP{JNOK1RJUW'T

f]a l aNPRVdWabWMJHSabf_Li/a l WJLOJqJRJHSab\pNO^'Td

r : U R2 R3 nL l N U rIJHbi/abVPRc^a`i JVPWJVPRTzV|WJJS l T'^aRaQCT(u1, u2)

n`'S'T'U'NPQ {J{VONuveRJU'WavqsPRJK1JHbi/aVuRT{NOWRJS'g

r1 =ru1

Tr2 =

ru2

i T'^'NPK'^'J^NPf]ab{T'VOTQCg%j P3xu]buuGIO-'mP]h'PPuubPu]m]PuPu3.>IuuOxPu

ut%uZ cO@hb$ .&n(WabWXT {MViyUaNqW'ST'{g(vnhJfx^'aU'aNPRnhURJ{J_{VONuvRJU'Wav^abwNOKe`J{NOSv^'JVuRTeVuyopNOVuR{yNPR`'Jbi^'J'NP^'^'JNWabVxabRNui c^JN`'S'JVuRSa^VPR{J'j%QCNP^'^'JGsPRJT_JHNOVP`'NOU'T{aFmRGyVi J{Td_RT'`a F

x36= 0 T|i T'^'NPKr^abd^'NPfxa{T'VPT'QCJVuRc{NOWRJS'J{

r1Tr2j

IJR]d {VONqRS'T J`'S'N l NuiNO^'Td sOW'{T'{ai NP^R^g%nIQCg{ l aic^'NOKwpNOQ Hby l NOQ`'Jbi cfOJ{abRcVd~RS'NuRcTQhnTVO`'JbicfOyFmoT'Qzi JWai c^'ghNWJJS l T'^'abRg%j:RJy l JHbr^'J~`'S'TSaVPVOQCJRS'NO^'TTW'S'T{g(v^a|`'J{NOSv^'JVuRTr^'N%^a l J~fxa l a{abRc|W'ST'{yF{

R3T_^aWi/a l gh{abRc^aG^'NONma^aiTRT'UNOVOWT'NIyVi J{Td-nURJHgXJ^ai NP\|ai/a^a

`'J{NPSv^'JVPRTna l JVuRabRJU'^'Jfxa l abRc_W'ST'{yF

: [a, b] U R2WabWLi a l WJNJRJHSab\|NP^'T'Nh{%JHi/aVPRc

Uj}Vi T~W'S'T'{abd|^'N(i Nu\TR{%JW'S'NPVPR^Jbr

VPRT{pJ l ^'TQCTTRNOQCT\|NmiJWai c^'ghQCTWJJS l T^abRaQCT-n'RJ~J^afxa l aNuRVdWaWVONOQCNPK'VPR{JJbRJHS'ab\|NO^T'K{eRaW'T'NJHbi/abVPRTj:RTJRJHSab\|NP^'TdVPWi NOT{aFmRrVd{JHi/aVPR]dvnWJRJS'ghNJ`'TVOgh{aFtRVd^'NPVOWJbi cW'T'QCTVOTVPRNPQaQCTWJJS l T'^aRnURJHg WJS'SNOWR^'J1fxa l abRc[W'S'T'{yF TNPN{NOWRJS'g VOWJS'JVPRTjegHy l NOQ f]ar^'T'QabRcVdi JWai c^JKRNPJS'T'NPK`'J{NOSv^'JVuRNOKn/`JsPRJQyQCJ]\^JpVOU'TRabRc'n URJ`'J{NPSv^'JVPRc_f]a l abNPRVd`'S'T`'JQCJbopTJ l ^'JKVOT'VuRNOQCgWJJS l T'^abRe WaSRgtuj

fGJ`'S'N l NuiNO^'TdWaVxaRNuic^'JLPJ`'S'JVPRS'a^'VPR{aTRS'NPHJ{ab^'TdS'NOLuyi'd'S'^'JrVPRTVuiN l yNPRnCURJ1{qWab\ l JKzRJU'WN

x = r(u1, u2)S'NOLuyi'd'S'^'JK`J{NOSv^'JVuRT

{NPWRJSg

r1Tr2

f]a l aFmRHabfOT'V%{Tx

j :RJR_HabfOT'V%^abfOgh{aNuRVd1 ]n].P%jLy%S'JQCN|RJLPJ'n-^aWaVOabRNi c^'JQ `'S'JVuRSa^VPR{N

Txn=SaVPVOQabRS'T'{abNOQCJQ

WabW1i T'^NOK'^'JN`'S'JVPRS'a^'VPR{J'nWJS'S'NPWR^'JJ`'S'N l Nui NP^a^'NP{ghS'J]\ l NO^'^'abd-nZ`'Jbri J]\TRNui c^'J~J`'S'N l Nui NP^'^abdW'{a l SabRT'U'^adJSQa

gnQabRS'T'a

GWJRJS'JK{

Wab^'J^'T'UNOVOWJQXHabfxT'VPNT'QCNONuR{T l

G =

(

r1, r1 r1, r2r1, r2 r2, r2

)

L l NUNOS'Nuf , JHJfO^a]U'NO^'J~VuRa^ l aSR^'JNNO{WiT l J{J~`'SJTfx{N l NP^'T'N{ R3 j 3' Z y%{a l SabRT'U'^abd"GJS'Qag^abfOgh{aNuRVd #b%i #bbC

&|n] %ikj%iq`'J{NPSv^'JVPRTj%NOS{abdW'{a l SabRT'U'^abdJSQa~`J{NOSv^'JVuRTe`Jfx{Jid'NuR_{ghU'T'Vui'dRc l i Tr

^'gT[yLigQCNP\ l yq{NPWRJS'aQCT{WaVxabRNui c^'JQ `'SJVPRSa^'VuR{N[{VOTiyqRJLOJnURJJ^'JJ`S'N l Nid'NuRRaQXVPWaid'S'^JN`'SJTfx{N l NP^'T'N]n l iT'^'gW'S'T{g(ve^'a_`'Jbr{NOSv^'JVuRTeT`i Jopa l cVxaQCJK`J{NOSv^'JVuRT-j

yVuRc v, w Tx {NOWRJSa|{|WaVxabRNui c^'JQ`S'JVPRSa^'VuR{NGT v,w ryLPJbiQCNP\ l y^'T'QCTn'RJL l a

|v| =

G(v, v), cosv,w =G(v, w)

|v| |w|`S'T

v 6= 0 T w 6= 0

yVuRc : [a, b] U S'NPLPy]id'S^abdW'S'T'{ad^aq`'J{NPSv^'JVPRT T =d

dt= (u1, u2)

rNONI{NOWRJSVOWJS'JVPRT{Gi JWai c^g(vWJJS l T^abRavnRJL l a l i T^aW'S'T'{JK

L() =

b

a

G(, ) dt,

}CVui T V U n'RJ_`i Jo|a l c_U'aVPRT r(V ) `J{NOSv^'JVuRT Sab{^a

Sr(V ) =

V

det G du1du2

1 l T'NOSNO^''T'ai c^'JKLONPJQCNPRS'T'T`NOS'{yF W'{a l S'abRT'U^yF JS'Qy~U'aVPRJ^arfxgh{aFtR #bA&~Pn&|[u/ |`'J{NOSv^'JVuRT-j1sPRJQViyUaNT'Vr`'Jbi cfPyFmRRaW[^abfOgh{aNOQyF ufxa`'T'VPc{ l T'NOS'NP^''Tai/av-bjZ%S'TqsPRJQ VxaQyGJS'Qy

gJHJfO^a]UaFmRpU'NOS'Nuf

ds2rhPW'{a l SabRpsui NPQCNO^Ra l i T^'gGbn'T-n{pWJJS l Tr

^abRav(u1, u2)

nfxa`'T'VPgh{aFtR{{T l N ds2 = G11(du

1)2 + G12(du1du2 + du2du1) + G22(du

2)2

}Vi T Li/a l WJN[QC^'JLPJJHS'abfxT'Nd'{bidNPRVd JHJHoNO^'T'NPQ `J^dRTd9S'NPLPy]id'S^'JK`'J{NPSv^'JVPRTn6RJa^ai JLOJQ`'NOS'{JKXW'{a l SabRTU'^'JKXJS'QCg `'J{NOSv^'JVuRT9{VuiyUaNGQC^'JLPJJHS'abfxT'Kd'{id'NuRVd b #b|.&Ij

\ bA]^a`pa^[LONuiT'WJT l r = (u sin, u cos, )

jfmaVPVOQCJRS'T'Q W'ST'{Jbri T'^'NPK'^'ghKRS'NPyLOJi c^'T'W^a|LONuiT'WJT l N

0 u sinh n 0 0 j GabKRT-aj`i Jopa l cW'S'T'{Ji T'^'NPK'^'JLPJ_RSNPyLOJbic^'T'Wa6 j l i T'^'g VPRJS'J^suRJLPJRS'NPyLPJbi c^'TWa 7 j'yLig sPRJLOJ~RS'NuyLOJbi c^'T'Wajc3'd~P ` aK l NOQ`'NOS'{byFW'{a l SabRT'U'^yFGJS'QyLONi T'WJT l aj

r1 =r

u= (sin, cos, 0), r2 =

r

= (u cos,u sin, 1)

JsPRJQyQabRS'Ta`NOS'{JKW'{a l SabRT'U'^'JKJS'QCg T'QCNONuR{T l

G =

(

1 00 u2 + 1

)

BCbCuu'ZO(bPuPuP(uuPbdu1

du2

]R>==x=u~]uuPObuTx

uPx3bu.>uuPPbPT x

bP'>uL>%Obuu 1%xuuxP'uAu6u=(Ids2

'hPPbp]uObu'>u'pPxb%''>OO'ub=%uub*>M>-%u-P'ds2

buPuuPvwu=umu

ds2(v, w) = G11v1w1 + G12(v

1w2 + v2w1) + G22v2w2

http : //mathworld.wolfram.com/Helicoid.htmlhttp : //www.coolphysics.com/4d/minimal/helicoid/helicoid.htmhttp : //www.xahlee.org/surface/helicoid catenoid/helicoid catenoidlg1.html

2 j Ii Jo|a l c~W'S'T'{Ji T'^'NPK'^'JLPJ_RSNPyLOJbic^'T'WaSab{^a

S =

0

0

sinh

0

u2 + 1 dud = [

fxaQCNO^'au = sinh x] =

20 + sinh2 0

4

6 j:RJS'J^g WS'T'{JbiT'^'NOK^'JLOJRS'NuyLPJbi c^T'Wa~^'aLONi T'WJT l NIf]a l aFmRVdVui N l yrFmopTQCTW'ST'{ghQCT ^aST'VPyKRNW'ST'{Jbi T^'NOK'^ghKRS'NPyLOJi c^'T'W^ap`i JVPWJVPRT-nVOJJR{NPRVPR{yFmopTK1JHbi/aVuRTTfxQCNP^'NO^'Tdi JWai c^'g(vWJJS l T^abR

1 : t [0, 0] 7 (0, t), 1 = (0, 1)2 : t [0, sin0] 7 (t, 0), 2 = (1, 0)

3 : t [0, 0] 7 (sinh t, t), 3 = (cosh t, 1)1(ghU'T'Vi T'Q l iT'^'g suRTvW'ST'{g(vj

L1 =

0

0

(0, 1)

(

1 00 1

) (

01

)

dt = 0

L2 =

sinh 0

0

(1, 0)

(

1 00 1 + t2

) (

10

)

dt = sinh0

L3 =

0

0

(cosh t, 1)

(

1 00 1 + sinh2 t

) (

cosh t1

)

dt

=

0

0

1 + cosh2 t+ sinh2 tdt =

0

0

2 cosh2 tdt =

2 sinh0

7 jf1(NPSwpT^'gRS'NPyLPJbi c^'TWaT'QCNPFmRVi N l yFtopT'Npi JWaic^'ghNpWJJS l T^abRgA12 = (0, 0)

nA13 = (0, 0)

nA23 = (sinh0, 0)

jyaVOabRNi c^'ghN{NPWRJSg WVPRJS'J^aQRS'NuyLPJbi c^T'Wa{~sPRTvRJU'WavTQCNOFmR_{T l

1(A12) = (0,1), 2(A23) = (1, 0), 3(A13) = (1, 1)

1(A13) = (0, 1), 2(A12) = (1, 0), 3(A23) = ( cosh0,1)NO`'NPS'c~QCJ\_^'J~{ghU'TVui TRcWJVPT'^yVPg yLi J{_RS'NPyLPJbi c^'TWa

cos12 =

(0,1)(

1 00 1

) (

10

)

(0,1)(

1 00 1

) (

01

)

(1, 0)

(

1 00 1

) (

10

)

= 0

%0

cos13 =

(1, 1)

(

1 00 1

) (

01

)

(1, 1)

(

1 00 1

) (

11

)

(0, 1)

(

1 00 1

) (

01

)

=12

cos23 =

(1, 0)(

1 00 1 + sinh2 0

) (

cosh01

)

(1, 0)(

1 00 1 + sinh2 0

) (

10

)

1

( cosh0,1)(

1 00 1 + sinh2 0

) (

cosh01

)

=12

k RVPF l aVi N l yNPRnURJyLi g9`'S'T{NOSwT'^avA12

nA13

nA23

Sa{^'gVOJJR{NPRrVPR{NO^'^'J

/2, /4, /4j1 l TGNPS'NO^'Tai c^JK~LONOJQCNPRST'T`'J{NOSv^'JVuRNOKVuiJ\_Ti/abVOc%VuiN l yFmopabd

RNOSQCT'^'Jbi JLOTdj k HbYZNOWRg TM{Ni T'U'T^'g%nJ`'SN l Nid'NPQCghN`NOS'{JKMW'{a l S'abRT'Ur^'JKXJS'QCJKX`S'T'^dRJM^abfOgh{abRcsi NOQCNP^RaQCT#bnha&I/u .i O.&Ijy{^yRS'NP^'^'NPKpLONPJQCNPRS'T'T-n^a`S'T'QCNOSnx`'S'T'^'a l i Nu\~abR l i T'^gW'S'T{g(vnxyLig@QCNP\~rl yW'S'T{ghQCT-n`i Jbo|a l T`'J{NOSv^'JVPRNOK-j ktl ^aWJ`'NOS{abdW'{a l SabRT'U'^abdJSrQa~J`'S'N l NuidNPR^'N%{VPNja`'ST'QCNOS-n{pRNOS'QCT^av`'NPS'{JKW'{a l SabRT'U'^'JKJSrQCg^'NuicfOdz{ghU'T'Vi TRceW'S'T'{Tfx^y[WS'T'{JKz^a`'J{NPSv^'JVPRTj k HYZNPWRgTz{Nuri T'U'T'^g%n l i'dJ`'S'N l Nui NP^'Td"WJRJS'g(v^'N l JVuRabRJU^'JM`'NOS{JKW'{a l S'abRT'U^'JKGJS'QCg%n^afxgh{aFtR|suiNOQCNO^RaQCT#b .d| 'iPG'&I `'J{NPSv^'JVPRTj:i'dTvJ`'T'VOa^'Td^aQ`'JRS'NOHyNPRVdq{RJSabdW'{a l SabRT'U'^abdJS'Qa~`'J{NOSv^'JVPRT-j

k HJfO^a]U'T'Q"UNOS'Nufn(u1, u2)

RabWyF^'JS'Qai cWSNOLPy]idS'^'JKq`J{NOSv^'JVuRT{RJU'WN

r(u1, u2)nURJ

(r1, r2, n)r`Jbi J]\TRNuic^'JJS'T'NO^RT'S'J{a^'^'ghKSNO`'NPS{

R3j k ^a~^avJ l TRVdd'{^J`JJSQyiN

n =r1 r2

||r1 r2||k HJfx^a]U'TQXU'NOS'Nuf

rjk{NPWRJS

2r/ujukj-a l a l T'Q"{`'S'JTfO{Jbi c^'JKRJU'WN

x VOT'QCQCNuRS'T'U^yFQabRS'T'y H n'`'JbiJ\_T'{

Hjk = rjk, nk WabfOgh{aNuRVd-nURJsuRTqVOJJR^'JbwpNO^TdqWJS'S'NPWR^JJ`'S'N l NuidFmR{`S'JTfO{Jbi cbr^'JKqRJUWN

x`'J{NPSv^'JVPRT

^NOWJRJSyF W'{a l SabRT'U'^yFJS'Qy

h^aWaVOar

RNuic^'JQ"`S'JVPRSa^'VuR{NTx

F j POpuu |.>u'%xuuPb

v1 = v1(u1, u2)

3' Z y{a l SabRT'U'^adGJS'Qah^abfxgh{aNPRVd #*&~/%i#bbC

&|n] %ikj%iq`'J{NPSv^'JVPRTj1MJRxi TU'T'NJRG`'NOS{JK~W'{a l SabRTU'^'JKJS'QCg`'J{NPSv^'JVPRTn{bRJSad~W'{a l r

SabRTU'^abdGJS'Qan{JJHopNLOJ{JSd-n^'NtJH]dfxa^aHg6Rc^'T^'NP{ghS'J]\ l NO^'^JK-n^'T`'Jbi J]\_TRNi c^'J_J`'S'N l Nui NP^'^'JKj

1(g6dVO^'TQ@LONPJQCNPRS'T'U'NPVOW'T'KVPQCghVui{bRJS'JK_W'{a l SabRT'U'^'JKJS'QCg%jyVuRc

: [a, b] U R2 r R3HT'S'NPLPy]id'S^abdW'S'T'{ad ^a|`'J{NPSv^'JVPRTj

r s tu Z y%S'T{Tfx^'aWS'T'{JK^a%`'J{NPSv^'JVPRTy l J{bi NuR{JSdNPR

yS'a{^'NP^'T'Fk cos =

H(, )

G(, ),

L l NryLOJieQCNu\ l y

nTnn'^'JS'Qaid'QCTWWS'T'{JKTW`J{NOSv^'JVuRT-j

yVuRc_{NOWRJSvr:WabVxabRNui c^ghK{NPWRJSW`J{NOSv^'JVuRTe{~RJU'WN

r(u1, u2)TnrI{NOWRJSM^'JS'Qai T@W`'J{NPSv^'JVPRT{sPRJK@\|NRJU'WNjSJ{N l NOQhnU'NOS'Nuf

RJU'Wyr(u1, u2)

l {byQCNPS'^yF`i JVPWJVPRc'nZ^abR]d'^yRbyF ^a{NPWRJS'gnTvj=%Nr

S'NOVPNOU'NP^'T'NqsPRJKX`i JVOWJVPRT9T"`'J{NPSv^JVPRTr|W'S'T{abdu1,u2,v

r^abfOgh{aNuRVd %xO']u n JR{NOUabFmopTQ9RJU'WN

r(u1, u2)TWaVxabRNui c^'JQy{NOWr

RJSyvjfRNPJS'NPQCg eNO^cN{g6RNOWaNuR

tLZ y%S'T'{Tfx^a~^'JSQai c^'JLOJVONOUNO^'Tdu1,u2,v

Sab{^a

kn(v) = H(v, v)

G(v, v)+9NPVui T^'JS'QaiTeWe`'J{NOSv^'JVPRTqTeWe^'JS'Qai c^JQyVONPU'NO^'TFVOJ{`'a l aFtRn

3T'^a]U'NbjaW'T'Q JHSabfOJQhnVOS'N l TX{VPNuvW'S'T'{g(v`S'JxvJ l doTvXUNOS'Nuf l a^'^yF RJUrWy`'J{NPSv^JVPRTeTT'QCNOFtopTv l a^^'ghKWaVOabRNi c^'ghK{NOWRJS

vn^'aT'QCNO^cwyF

W'S'T'{Tfx^yHy l NuRTQCNPRc_^'JS'Qai c^'JN%VPNOU'NP^'T'Nbj%QCNONuRQCNPVPRJ_VPJJR^JwpNP^'T'N

k(v)| cos | = kn(v)^JL l a~T'QCNO^'^J_suRJ_yR{NOS\ l NO^'TN^abfOgh{aNuRRNOJSNOQCJK eNO^'cNjv2 = v2(u1, u2)

==Obb'>Cub'PbZ']PC(H(u) = JT H(v(u))J

Ju=xuPmu'xuPuM?CuP''

uObPuu'uO ||(l)|| = 1 hbP : [a, b] R3b=OuVI@REUTO%Y'G*]

d2/dl2 6= 0GAWOJRC.u'>Iu'>Iu=Puumuu

n =d2

dl2/||d2

dl2||

CI@IZXG* PbP.>Au'>Iu=Puumub k = ||d2

dl2||

x

\ bA](`qmaVOVPQCJRST'Q `'Sd'QCJK W'SyLOJ{JK 'Ti T'^ l S-jm%yVPRc

r`i JVrWJVPRcn`NOS'NPVONOWaFto|abdJVOc'TiT'^ l Sa`'J l yLi JQ

j:Ra`i JVPWJVPRc`'NOSNOVONr

WabNPReTi T'^ l S@`'J^'NOWJRJS'JQy1suii T'`'VuyjGabKRTzW'S'T'{Tfx^y1sPRJLOJsui'iT'`'Vxa{RJU'WN

pn^abT'HJbi NPN%y l ai NO^^'JKJRNOLOJ'NP^RSaj

c3'd~P `=j(`'JVOJH'j-a l a l T'Qq'Ti T'^ l S|`'aSaQCNuRS'T'UNOVOWT_`'ST|`'JQCJoTi JWai c^g(v|WJJS l T'^'abR

(, z)

r = (A cos,A sin, z)1(ghU'T'Vi T'Q"`'NPS'{yFT{RJSyF W'{a l SabRTU'^'ghNJSQCg 'TiT'^ l Sa

r1 =r

= (A sin,A cos, 0), r2 =

r

z= (0, 0, 1)

G =

(

A2 00 1

)

n =r1 r2

||r1 r2||= (cos, sin, 0)

r11 =2r

2= (A cos,A sin, 0), r12 =

2r

z= 0, r22 =

2r

z2= 0

H =

(

A 00 0

)

yGabVxabRNui c^ghK{NOWRJS_Wsuii T'`'Vuy_{%RJU'WNpT'QCNPNPRp{T l

(p) = (, 0)jaWWabW

^'JS'QaicW`J{NOSv^'JVuRTM^ab`'Sa{biNO^a{J{^'NPw^'JVPRce'TiT'^ l Sana^'JS'Qai cWsi'i T'`'Vuyr3{^yRSc'nRJ~yLPJbi

= /2 + j%JsuRJQy

k(p) cos (/2 + ) =

(, 0)

(

A 00 0

) (

0

)

(, 0)

(

A2 00 1

) (

0

) = 1A

k(p) =1

A sin (`'JVPJH'jabWWaW^JS'Qai c^'ghQCTVPNOU'NP^'Td'QCTqTi T'^ l Sa_d'{id'FtRVdeJW'Sy\r^'JVPRT W'ST'{TfO^'gWJRJSg(v@QCgfO^aNPQ3n=RJy l JH^'J{JVO`Jbi cfOJ{abRcVd@JSrQyiJK

k| cos | = kn(v)1 l a^'^JQ VuiyUaNvrIWaVxaRNuic^'ghK{NOWRJSMW@si'i T'`VPy@{RJUWN

pnkn(v) =

1/AnZRj W@`iJVOWJVuRc'n^abR]d'^yRadM^a{NOWRJSa

nTv`'NPS'NOVPNOWaNuR['Ti T'^ l S@`'J

JW'Sy\_^'JVPRT-n = /2+

j1TRJLONI`JbiyUaNPQRJR\pNmJR{NPRk(p) = 1

A sin

j

\ bA]`aKRT|`'NPS'{yF"T{bRJSyF"W{a l SaRT'U'^ghNhGJS'QCgM`'J{NPSv^JVPRT-nf]a l ab^'^'JK{_{T l N%LOS'aT'Wa_%y^'W''T'T

z = f(x, y)c3'd~P `aWabdM`'J{NOSv^'JVuRcJU'NO^'c1`'S'JVPRJ[f]a l aNPRVd{[`abSaQCNPRS'T'U'NrVOWJQ{T l N

r(u1, u2) = (u1, u2, f(u1, u2))j:pai NON{VPN`'J~J`'S'N l Nui NP^'T'Fj

r1 =r

u1= (1, 0, f1), r2 =

r

u2= (0, 1, f2)

G =

(

1 + f 21 f12f12 1 + f

22

)

n =(f1,f2, 1)

1 + f 21 + f22

r11 = (0, 0, f11), r12 = (0, 0, f12), r22 = (0, 0, f22)

H =1

1 + f 21 + f22

(

f11 f12f12 f22

)

\ bA]`%JVPRS'JT'Q`'Jy^'W'T'Ty = f(x)

`'J{NPSv^'JVPRc~{S'abopNP^'Tdz `Jbr{SabopaNOQ@LPSaT'W_{JW'SyLtJVOT

OxjaKRT_`'NOS{yF T_{RJSyF W'{a l SabRT'U'^'ghN

GJS'QCg suRJK`'J{NPSv^'JVPRTjc3'd~P ` %J{NOSv^'JVPRc{S'abopNP^'TdQCJ]\^'J|fxa l abRc_`abSaQCNPRS'T'U'NPVOW'T

r(u, ) = (u, f(u) cos, f(u) sin)

|ai NON{VPNG`J_J`S'N l Ni NO^T'Fj

r1 =r

u= (1, f (u) cos, f (u) sin), r2 =

r

= (0,f(u) sin, f(u) cos)

G =

(

1 + (f )2 00 f 2

)

n =(f , cos, sin)

1 + (f )2

r11 = (0, f(u) cos, f (u) sin), r12 = (0,f (u) sin, f (u) cos),

r22 = (0,f(u) cos,f(u) sin)

H =1

1 + (f )2

(

f 00 f

)

#/$%bzB' 2\ bA]a`GabK l TRNyLOJi"QCNu\ l yzi T'^'TdQCT

v = u + 1Tv = 3 u ^'a`'J{NPSv^'JVPRT

x = u cos v, y = u sin v, z = u2

\ bA]C`1:ghU'T'VuiTRc{bRJSyF W'{a l SabRT'U'^yFJSQyRJS'aj