pdfsharp c# : Add links in pdf SDK software API .net windows asp.net sharepoint TRENCH_REAL_ANALYSIS41-part261

402 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Everypointintheuv-planecanbewritteninpolarcoordinatesas
uDcos˛; vDsin˛;
whereeitherD0or
D
p
u2Cv>0; ˛<;
andthepointsforwhichD0or˛Dareoftheform.u;0/,withu0(Figure6.3.5).
If.u;v/DF.x;y/forsome.x;y/inS,then(6.3.15)impliesthat>0and <˛<
. Conversely, , anypointintheuv-planewithpolarcoordinates.;˛/satisfyingthese
conditionsistheimageunderFofthepoint
.x;y/D.
1=2
cos˛=2;
1=2
sin˛=2/2S:
Thus,
F
1
S
.u;v/D
2
4
.u
2
Cv
2
/
1=4
cos.arg.u;v/=2/
.uCv2/1=4sin.arg.u;v/=2
3
5
; <arg.u;v/<:
v
u
(u,v)
α
α = −π
u2 + v2
Figure6.3.5
Becauseof(6.3.8),Falsomapstheopenlefthalf-plane
S
1
D
˚
.x;y/
ˇ
ˇ
x<0
ontoF.S/,and
F
1
S
1
.u;v/D
2
4
.u2Cv2/1=4cos.arg.u;v/=2/
.u
2
Cv
2
/
1=4
sin.arg.u;v/=2/
3
5
;   <arg.u;v/<3;
DF
1
S
.u;v/:
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Section6.3
TheInverseFunctionTheorem
403
Example6.3.5
Thetransformation
u
v
DF.x;y/D
e
x
cosy
e
x
siny
(6.3.16)
isnotone-to-one,since
F.x;yC2k/DF.x;y/
(6.3.17)
ifkisanyinteger. Thistransformationisone-to-oneonasetSifandonlyifSdoesnot
containanypairofpoints.x
0
;y
0
/and.x
0
;y
0
C2k/,wherekisanonzerointeger.This
conditionisnecessarybecauseof(6.3.17);weleaveittoyoutoshowthatitissufficient
(Exercise6.3.8).Therefore,forexample,Fisone-to-oneon
S
D
˚
.x;y/
ˇ
ˇ
1<x<1;y<C2
(6.3.18)
whereisarbitrary.Geometrically,S
istheinfinitestripboundedbythelinesyDand
yDC2.ThelowerboundaryisinS
,buttheupperisnot(Figure6.3.6).Sinceevery
pointisintheinteriorofsomesuchstrip,Fislocallyinvertibleontheentireplane.
y
x
y = φ
y = φ + 2π
Figure6.3.6
TherangeofF
S
istheentireuv-planeexcepttheorigin,sinceif.u;v/¤.0;0/,then
.u;v/canbewrittenuniquelyas
u
v
D
cos˛
sin˛
;
where
>0; ˛<C2;
so.u;v/istheimageunderFof
.x;y/D.log;˛/2S:
TheoriginisnotinR.F/,since
jF.x;y/j
2
D.e
x
cosy/
2
C.e
x
siny/
2
De
2x
¤0:
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404 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Finally,
F
1
S
.u;v/D
2
4
log.u
2
Cv
2
/
1=2
arg.u;v/
3
5
; arg.u;v/<C2:
ThedomainofF
1
S
istheentireuv-planeexceptfor.0;0/.
RegularTransformations
ThequestionofinvertibilityofanarbitrarytransformationFWR
n
!R
n
istoogeneralto
haveausefulanswer.However,thereisausefulandeasilyapplicablesufficientcondition
whichimpliesthatone-to-onerestrictionsofcontinuouslydifferentiabletransformations
havecontinuouslydifferentiableinverses.
Tomotivateourstudyofthisquestion,letusfirstconsiderthelineartransformation
F.X/DAXD
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
:
:
:
:
:
:
:
:
:
:
:
:
a
n1
a
n2
 a
nn
3
7
7
7
5
2
6
6
6
4
x
1
x
2
:
:
:
x
n
3
7
7
7
5
:
FromTheorem6.3.1,FisinvertibleifandonlyifAisnonsingular,inwhichcaseR.F/D
R
n
and
F
1
.U/DA
1
U:
SinceAandA
1
arethedifferentialmatricesofFandF
1
,respectively,wecansaythata
lineartransformationisinvertibleifandonlyifitsdifferentialmatrixF
0
isnonsingular,in
whichcasethedifferentialmatrixofF
1
isgivenby
.F
1
/
0
D.F
0
/
1
:
Becauseofthis,itistemptingtoconjecturethatifFWR
n
!R
n
iscontinuouslydifferen-
tiableandA
0
.X/isnonsingular,or,equivalently,JF.X/¤0,forXinasetS,thenFis
one-to-oneonS.However,thisisfalse.Forexample,if
F.x;y/D
e
x
cosy
e
x
siny
;
then
JF.x;y/D
ˇ
ˇ
ˇ
ˇ
e
x
cosy e
x
siny
e
x
siny
e
x
cosy
ˇ
ˇ
ˇ
ˇ
De
2x
¤0;
(6.3.19)
butFisnotone-to-oneonR
2
(Example6.3.5). Thebestthatcanbesaidingeneralis
thatifFiscontinuouslydifferentiableandJF.X/¤0inanopensetS,thenFislocally
invertibleonS,andthelocalinversesarecontinuouslydifferentiable. Thisispartofthe
inversefunctiontheorem,whichwe willprovepresently. First, , weneedthefollowing
definition.
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Section6.3
TheInverseFunctionTheorem
405
Definition6.3.2
AtransformationFWR
n
!R
n
isregularonanopensetS ifFis
one-to-oneandcontinuouslydifferentiableonS,andJF.X/¤0ifX2S. Wewillalso
saythatFisregularonanarbitrarysetSifFisregularonanopensetcontainingS.
Example6.3.6
If
F.x;y/D
xy
xCy
(Example6.3.2),then
JF.x;y/D
ˇ
ˇ
ˇ
ˇ
1 1
1
1
ˇ
ˇ
ˇ
ˇ
D2;
soFisone-to-oneonR
2
.Hence,FisregularonR
2
.
If
F.x;y/D
xC y
2xC2y
(Example6.3.3),then
JF.x;y/D
ˇ
ˇ
ˇ
ˇ
1 1
2 2
ˇ
ˇ
ˇ
ˇ
D0;
soFisnotregularonanysubsetofR
2
.
If
F.x;y/D
x
2
y
2
2xy
(Example6.3.4),then
JF.x;y/D
ˇ
ˇ
ˇ
ˇ
2x 2y
2y
2x
ˇ
ˇ
ˇ
ˇ
D2.x
2
Cy
2
/;
soFisregularonanyopensetSonwhichFisone-to-one,providedthat.0;0/62S.Forex-
ample,Fisregularontheopenhalf-plane
˚
.x;y/
ˇ
ˇ
x>0
,sincewesawinExample6.3.4
thatFisone-to-oneonthishalf-plane.
If
F.x;y/D
e
x
cosy
e
x
cosy
(Example6.3.5),thenJF.x;y/ De
2x
(see(6.3.19)),soFisregularonanyopenseton
whichitisone-to-one.TheinteriorofS
in(6.3.18)isanexampleofsuchaset.
Theorem6.3.3
SupposethatF W R
n
! R
n
isregular onanopensetS;andlet
GDF
1
S
:ThenF.S/isopen;GiscontinuouslydifferentiableonF.S/;and
G
0
.U/D.F
0
.X//
1
; where UDF.X/:
Moreover;sinceGisone-to-oneonF.S/;GisregularonF.S/:
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406 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Proof
WefirstshowthatifX
0
2 S,thenaneighborhoodofF.X
0
/isinF.S/. This
impliesthatF.S/isopen.
SinceS isopen,thereisa > 0suchthat
B
.X
0
/  S. LetBbetheboundaryof
B
.X
0
/;thus,
BD
˚
ˇ
ˇ
X
jXX
0
jD:
(6.3.20)
Thefunction
.X/DjF.X/F.X
0
/j
iscontinuousonSandthereforeonB,whichiscompact.Hence,byTheorem5.2.12,there
isapointX
1
inBwhere.X/attainsitsminimumvalue,saym,onB.Moreover,m>0,
sinceX
1
¤X
0
andFisone-to-oneonS.Therefore,
jF.X/F.X
0
/jm>0 if jXX
0
jD:
(6.3.21)
Theset
˚
U
ˇ
ˇ
jUF.X
0
/j<m=2
isaneighborhoodofF.X
0
/.WewillshowthatitisasubsetofF.S/. Toseethis,letUbe
afixedpointinthisset;thus,
jUF.X
0
/j<m=2:
(6.3.22)
Considerthefunction
1
.X/DjUF.X/j
2
;
whichiscontinuousonS.Notethat
1
.X/
m
2
4
if jXX
0
jD;
(6.3.23)
sinceifjXX
0
jD,then
jUF.X/jDj.UF.X
0
//C.F.X
0
/F.X//j
ˇ
ˇ
jF.X
0
/F.X/jjUF.X
0
/j
ˇ
ˇ
m
m
2
D
m
2
;
from(6.3.21)and(6.3.22).
Since
1
iscontinuousonS,
1
attainsaminimumvalueonthecompactset
B
.X
0
/
(Theorem5.2.12);thatis,thereisan
Xin
B
.X
0
/suchthat
1
.X/
1
.
X/D; X2
B
.X
0
/:
SettingXDX
0
,weconcludefromthisand(6.3.22)that
1
.
X/D
1
.X
0
/<
m
2
4
:
Becauseof(6.3.20)and(6.3.23),thisrulesoutthepossibilitythat
X2B,so
X2B
.X
0
/.
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Section6.3
TheInverseFunctionTheorem
407
NowwewanttoshowthatD0;thatis,UDF.
X/. Tothisend,wenotethat
1
.X/
canbewrittenas
1
.X/D
Xn
jD1
.u
j
f
j
.X//
2
;
so
1
isdifferentiableonB
p
.X
0
/. Therefore,thefirstpartialderivativesof
1
areallzero
atthelocalminimumpoint
X(Theorem5.3.11),so
Xn
jD1
@f
j
.
X/
@x
i
.u
j
f
j
.
X//D0; 1in;
or,inmatrixform,
F
0
.
X/.UF.
X//D0:
SinceF
0
.
X/isnonsingularthisimpliesthatUDF.
X/(Theorem6.1.13). Thus,wehave
shownthateveryUthatsatisfies(6.3.22)isinF.S/. Therefore,sinceX
0
isanarbitrary
pointofS,F.S/isopen.
Next,weshowthatGiscontinuousonF.S/. SupposethatU
0
2 F.S/andX
0
isthe
uniquepointinSsuchthatF.X
0
/DU
0
.SinceF
0
.X
0
/isinvertible,Lemma6.2.6implies
thatthereisa>0andanopenneighborhoodN ofX
0
suchthatN Sand
jF.X/F.X
0
/jjXX
0
j if X2N:
(6.3.24)
(Exercise6.2.18alsoimpliesthis.)SinceFsatisfiesthehypothesesofthepresenttheorem
onN,thefirstpartofthisproofshowsthatF.N/isanopensetcontainingU
0
DF.X
0
/.
Therefore,thereisaı>0suchthatXDG.U/isinNifU2B
ı
.U
0
/.SettingXDG.U/
andX
0
DG.U
0
/in(6.3.24)yields
jF.G.U//F.G.U
0
//jjG.U/G.U
0
/j if U2B
ı
.U
0
/:
SinceF.G.U//DU,thiscanberewrittenas
jG.U/G.U
0
/j
1
jUU
0
j if U2B
ı
.U
0
/;
(6.3.25)
whichmeansthatGiscontinuousatU
0
.SinceU
0
isanarbitrarypointinF.S/,itfollows
thatGiscontinousonF.S/.
WewillnowshowthatGisdifferentiableatU
0
.Since
G.F.X//DX; X2S;
thechainrule(Theorem6.2.8)impliesthatifGisdifferentiableatU
0
,then
G
0
.U
0
/F
0
.X
0
/DI
408 Chapter6
Vector-ValuedFunctionsofSeveralVariables
(Example6.2.3). Therefore,ifGisdifferentiableatU
0
,thedifferentialmatrixofGmust
be
G
0
.U
0
/DŒF
0
.X
0
/
1
;
sotoshowthatGisdifferentiableatU
0
,wemustshowthatif
H.U/D
G.U/G.U
0
/ŒF
0
.X
0
/
1
.UU
0
/
jUU
0
j
.U¤U
0
/;
(6.3.26)
then
lim
U!U
0
H.U/D0:
(6.3.27)
SinceFisone-to-oneonSandF.G.U//DU,itfollowsthatifU¤U
0
,thenG.U/¤
G.U
0
/.Therefore,wecanmultiplythenumeratoranddenominatorof(6.3.26)byjG.U/
G.U
0
/jtoobtain
H.U/D
jG.U/G.U
0
j
jUU
0
j
G.U/G.U
0
/ŒF
0
.X
0
/
1
.UU
0
/
jG.U/G.U
0
/j
!
D
jG.U/G.U
0
/j
jUU
0
j
F
0
.X
0
/
1
UU
0
F
0
.X
0
/.G.U/G.U
0
//
jG.U/G.U
0
/j
if0<jUU
0
j<ı.Becauseof(6.3.25),thisimpliesthat
jH.U/j
1
kŒF
0
.X
0
/
1
k
ˇ
ˇ
ˇ
ˇ
UU
0
F
0
.X
0
/.G.U/G.U
0
//
jG.U/G.U
0
/j
ˇ
ˇ
ˇ
ˇ
if0<jUU
0
j<ı.Nowlet
H
1
.U/D
UU
0
F
0
.X
0
/.G.U/G.U
0
//
jG.U/G.U
0
/j
Tocompletetheproofof(6.3.27),wemustshowthat
lim
U!U
0
H
1
.U/D0:
(6.3.28)
SinceFisdifferentiableatX
0
,weknowthatif
H
2
.X/D lim
X!X
0
F.X/F.X
0
/F
0
.X
0
/.XX
0
/
jXX
0
j
;
then
lim
X!X
0
H
2
.X/D0:
(6.3.29)
SinceF.G.U//DUandX
0
DG.U
0
/,
H
1
.U/DH
2
.G.U//:
Section6.3
TheInverseFunctionTheorem
409
Nowsupposethat>0.From(6.3.29),thereisaı
1
>0suchthat
jH
2
.X/j< if 0<jXX
0
jDjXG.U
0
/j<ı
1
:
(6.3.30)
SinceGiscontinuousatU
0
,thereisaı
2
2.0;ı/suchthat
jG.U/G.U
0
/j<ı
1
if 0<jUU
0
j<ı
2
:
Thisand(6.3.30)implythat
jH
1
.U/jDjH
2
.G.U//j< if 0<jUU
0
j<ı
2
:
Sincethisimplies(6.3.28),GisdifferentiableatX
0
.
SinceU
0
isanarbitrarymemberofF.N/, wecannowdropthezerosubscriptand
concludethatGiscontinuousanddifferentiableonF.N/,and
G
0
.U/DŒF
0
.X/
1
; U2F.N/:
ToseethatGiscontinuouslydifferentiableonF.N/,weobservethatbyTheorem6.1.14,
eachentryofG
0
.U/(thatis,eachpartialderivative@g
i
.U/=@u
j
,1  i;j   n)canbe
writtenastheratio,withnonzerodenominator,ofdeterminantswithentriesoftheform
@f
r
.G.U//
@x
s
:
(6.3.31)
Since@f
r
=@x
s
iscontinuousonN andGiscontinuousonF.N/,Theorem5.2.10implies
that(6.3.31)iscontinuousonF.N/. Sinceadeterminantisacontinuousfunctionofits
entries,itnowfollowsthattheentriesofG
0
.U/arecontinuousonF.N/.
Branches oftheInverse
IfFisregularonanopensetS,wesaythatF1
S
isabranchof F1. (Thisisaconvenient
terminologybutisnotmeanttoimplythatFactuallyhasaninverse.)Fromthisdefinition,
itispossibletodefineabranchofF
1
onasetT R.F/ifandonlyifTDF.S/,where
FisregularonS. TheremaybeopensubsetsofR.F/thatdonothavethisproperty,and
thereforenobranchofF
1
canbedefinedonthem. ItisalsopossiblethatT DF.S
1
/D
F.S
2
/,whereS
1
andS
2
aredistinctsubsetsofD
F
.Inthiscase,morethanonebranchof
F
1
isdefinedonT. Thus,wesawinExample6.3.4thattwobranchesofF
1
maybe
definedonasetT. InExample6.3.5infinitelymanybranchesofF
1
aredefinedonthe
sameset.
ItisusefultodefinebranchesoftheargumentTodothis,wethinkoftherelationship
betweenpolarandrectangularcoordinatesintermsofthetransformation
x
y
DF.r;/D
rcos
rsin
;
(6.3.32)
whereforthemomentweregardrandasrectangularcoordinatesofapointinanr-
plane.LetSbeanopensubsetoftherighthalfofthisplane(thatis,S
˚
.r;/
ˇ
ˇ
r>0
)
410 Chapter6
Vector-ValuedFunctionsofSeveralVariables
thatdoesnotcontainanypairofpoints.r;/and.r;C2k/,wherekisanonzerointeger.
ThenFisone-to-oneandcontinuouslydifferentiableonS,with
F
0
.r;/D
cos rsin
sin
rcos
(6.3.33)
and
JF.r;/Dr>0; .r;/2S:
(6.3.34)
Hence, FisregularonS. NowletT D D F.S/, thesetofpointsinthexy-planewith
polarcoordinatesinS. Theorem6.3.3statesthatT T isopenandF
S
hasacontinuously
differentiableinverse(whichwedenotebyG,ratherthanF
1
S
,fortypographicalreasons)
r
DG.x;y/D
2
4
p
x2Cy2
arg
S
.x;y/
3
5
; .x;y/2T;
wherearg
S
.x;y/istheuniquevalueofarg.x;y/suchthat
.r;/D
p
x2Cy2;arg
S
.x;y/
2S:
Wesaythatarg
S
.x;y/ isabranchoftheargumentdefinedonT. Theorem6.3.3also
impliesthat
G
0
.x;y/D
F
0
.r;/
1
D
"
cos
sin
sin
r
cos
r
#
(see(6.3.33))
D
2
6
4
x
p
x2Cy2
y
p
x2Cy2
y
x2Cy2
x
x2Cy2
3
7
5
(see(6.3.32)):
Therefore,
@arg
S
.x;y/
@x
D
y
x2Cy2
;
@arg
S
.x;y/
@y
D
x
x2Cy2
:
(6.3.35)
Abranchofarg.x;y/canbedefinedonanopensetT ofthexy-planeifandonlyif
thepolarcoordinatesofthepointsinT formanopensubsetofther-planethatdoesnot
intersectthe-axisorcontainanytwopointsoftheform.r;/and.r;C2k/,where
kisanonzerointeger. Nosubsetcontainingtheorigin.x;y/D.0;0/hasthisproperty,
nordoesanydeletedneighborhoodoftheorigin(Exercise6.3.14),sothereareopensets
onwhichnobranchoftheargumentcanbedefined.However,ifonebranchcanbedefined
onT,thensocaninfinitelymanyothers.(Why?)Allbranchesofarg.x;y/havethesame
partialderivatives,givenin(6.3.35).
Section6.3
TheInverseFunctionTheorem
411
Example6.3.7
Theset
T D
˚
.x;y/
ˇ
ˇ
.x;y/¤.x;0/ with x0
;
whichistheentirexy-planewiththenonnegativex-axisdeleted,canbewrittenasT D
F.S
k
/,whereFisasin(6.3.32),kisaninteger,and
S
k
D
˚
.r;/
ˇ
ˇ
r>0;2k<<2.kC1/
:
Foreachintegerk,wecandefineabrancharg
S
k
.x;y/oftheargumentinS
k
bytaking
arg
S
k
.x;y/tobethevalueofarg.x;y/thatsatisfies
2k<arg
S
k
.x;y/<2.kC1/:
EachofthesebranchesiscontinuouslydifferentiableinT,withderivativesasgivenin
(6.3.35),and
arg
S
k
.x;y/arg
S
j
.x;y/D2.kj/; .x;y/2T:
Example6.3.8
Returningtothetransformation
u
v
DF.x;y/D
x2y2
2xy
;
wenowseefromExample6.3.4thatabranchGofF
1
canbedefinedonanysubsetT of
theuv-planeonwhichabranchofarg.u;v/canbedefined,andGhastheform
x
y
DG.u;v/D
2
4
.u
2
Cv
2
/
1=4
cos.arg.u;v/=2/
.u
2
Cv
2
/
1=4
sin.arg.u;v/=2/
3
5
; .u;v/2T;
(6.3.36)
wherearg.u;v/isabranchoftheargumentdefinedonT. IfG
1
andG
2
aredifferent
branchesofF
1
definedonthesamesetT,thenG
1
D˙G
2
.(Why?)
FromTheorem6.3.3,
G
0
.u;v/D
F
0
.x;y/
1
D
2x 2y
2y
2x
1
D
1
2.xCy2/
x y
y x
:
Substitutingforxandyintermsofuandvfrom(6.3.36),wefindthat
@x
@u
D
@y
@v
D
x
2.x2Cy2/
D
1
2.uCv2/1=4
cos.arg.u;v/=2/
(6.3.37)
and
@x
@v
D
@y
@u
D
y
2.x2Cy2/
D
1
2.uCv2/1=4
sin.arg.u;v/=2/:
(6.3.38)
Itisessentialthatthesamebranchoftheargumentbeusedhereandin(6.3.36).
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