pdfsharp c# : Add hyperlink pdf file SDK software API .net windows asp.net sharepoint TRENCH_REAL_ANALYSIS40-part260

392 Chapter6
Vector-ValuedFunctionsofSeveralVariables
(b)
ShowthatM.A/DkAk.
(c)
Prove:Ifn>mornDmandAissingular,thenm.A/D0.(Thisrequiresa
resultfromlinearalgebraontheexistenceofnontrivialsolutionsofAXD0.)
(d)
Prove:IfnDmandAisnonsingular,then
m.A/M.A
1
/Dm.A
1
/M.A/D1:
10.
WesaythatFWR
n
!R
m
isuniformlycontinuousonSifeachofitscomponents
isuniformlycontinuousonS. Prove: IfFisuniformlycontinuousonS,thenfor
each>0thereisaı>0suchthat
jF.X/F.Y/j<
if jXYj<ı and X;Y2S:
11.
ShowthatifFiscontinuousonR
n
andF.XCY/DF.X/CF.Y/forallXandY
inR
n
,thenAislinear.H
INT
:Therationalnumbersaredenseinthereals:
12.
FindF
0
andJF.ThenfindanaffinetransformationGsuchthat
lim
X!X
0
F.X/G.Y/
XX
0
D0:
(a)
F.x;y;´/D
2
4
x
2
CyC2´
cos.xCyC´/
e
xy´
3
5
; X
0
D.1;1;0/
(b)
F.x;y/D
e
x
cosy
e
x
siny
; X
0
D.0;=2/
(c)
F.x;y;´/D
2
4
x
2
y
2
y
2
´
2
´
2
x
2
3
5
; X
0
D.1;1;1/
13.
FindF0.
(a)
F.x;y;´/D
.xCyC´/e
x
.x
2
Cy
2
/e
x
(b)
F.x/D
2
6
6
6
4
g
1
.x/
g
2
.x/
:
:
:
g
n
.x/
3
7
7
7
5
(c)
F.x;y;´/D
2
4
e
x
siny´
e
y
sinx´
e´sinxy
3
5
14.
FindFandJF.
(a)
F.r;/D
rcos
rsin
(b)
F.r;;/D
2
4
rcoscos
rsincos
rsin
3
5
(c)
F.r;;´/D
2
4
rcos
rsin
´
3
5
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Section6.2
ContinuityandDifferentiabilityofTransformations
393
15.
Prove:IfG
1
andG
2
areaffinetransformationsand
lim
X!X
0
G
1
.X/G
2
.Y/
jXX
0
j
D0;
thenG
1
DG
2
.
16.
ProveTheorem6.2.3.
17.
ShowthatifFWR
n
!R
m
isdifferentiableatX
0
and>0,thereisaı>0such
that
jF.X/F.X
0
/j.kF
0
.X
0
/kC/jXX
0
j if jXX
0
j<ı:
ComparethiswithLemma6.2.5.
18.
SupposethatFWR!RisdifferentiableatX
0
andF0.X
0
/isnonsingular.Let
rD
1
kŒF0.X
0
/1k
andsupposethat>0.Showthatthereisaı>0suchthat
jF.X/F.X
0
/j.r/jXX
0
j if jXX
0
j<ı:
ComparethiswithLemma6.2.6.
19.
Prove:IfLWR
n
!R
m
isdefinedbyL.X/DA.X/,whereAisnonsingular,then
jL.X/L.Y/j
1
kA1k
jXYj
forallXandYinR
n
.
20.
UseTheorem6.2.8tofindH
0
.U
0
/,whereH.U/DF.G.U/.Checkyourresultsby
expressingHdirectlyintermsofUanddifferentiating.
(a)
F.x;y;´/ D
2
4
x
2
Cy
2
´
x2Cy2
3
5
; G.u;v;w/ D
2
6
6
4
wcosusinv
wsinusinv
wcosv
3
7
7
5
, U
0
D
.=2;=2;2/
(b)
F.x;y/D
2
4
x
2
y
2
y
x
3
5
; G.u;v/D
"
vcosu
vsinu
#
; U
0
D.=4;3/
(c)
F.x;y;´/D
2
4
3xC4yC2´C6
4x2yC ´1
xC yC ´2
3
5
; G.u;v/D
2
4
u v
uC v
u2v
3
5
,
U
0
arbitrary
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394 Chapter6
Vector-ValuedFunctionsofSeveralVariables
(d)
F.x;y/D
xCy
xy
; G.u;v;w/D
2uvCw
e
u
2
v
2
; U
0
D.1;1;2/
(e)
F.x;y/D
x2Cy2
x2y2
; G.u;v/D
eucosv
eusinv
; U
0
D.0;0/
(f)
F.x;y/D
2
4
xC2y
xy
2
x
2
Cy
3
5
; G.u;v/D
uC2v
2uv
2
; U
0
D.1;2/
21.
SupposethatFandGarecontinuouslydifferentiableonRn,withvaluesinRn,and
letHDFıG.Showthat
@.h
1
;h
2
;:::;h
n
/
@.u
1
;u
2
;:::;u
n
/
D
@.f
1
;f
2
;:::;f
n
/
@.x
1
;x
2
;:::;x
n
/
@.g
1
;g
2
;:::;g
n
/
@.u
1
;u
2
;:::;u
n
/
:
WhereshouldtheseJacobiansbeevaluated?
22.
SupposethatF WR
n
!R
m
and
XisalimitpointofD
F
containedinD
F
. Show
thatFiscontinuousat
Xifandonlyiflim
k!1
F.X
k
/DF.
X/wheneverfX
k
gisa
sequenceofpointsinD
F
suchthatlim
k!1
X
k
D
X.H
INT
:SeeExercise5.2.15:
23.
SupposethatFWR
n
!R
m
iscontinuousonacompactsubsetSofR
n
.Showthat
F.S/isacompactsubsetofRm.
6.3THEINVERSEFUNCTIONTHEOREM
Sofarourdiscussionoftransformationshasdealtmainlywithpropertiesthatcouldjustas
wellbedefinedandstudiedbyconsideringthecomponentfunctionsindividually.Nowwe
turntoquestionsinvolvingatransformationasawhole,thatcannotbestudiedbyregarding
itasacollectionofindependentcomponentfunctions.
InthissectionwerestrictourattentiontotransformationsfromR
n
toitself.Itisuseful
tointerpretsuchtransformationsgeometrically. IfFD.f
1
;f
2
;:::;f
n
/,wecanthinkof
thecomponentsof
F.X/D.f
1
.X/;f
2
.X/;:::;f
n
.X//
asthecoordinatesofapointUDF.X/inanother“copy”ofR
n
.Thus,UD.u
1
;u
2
;:::;u
n
/,
with
u
1
Df
1
.X/; u
2
Df
2
.X/; :::; u
n
Df
n
.X/:
WesaythatFmapsXtoU,andthatUistheimageofXunder F. . Occasionallywewill
alsowrite@u
i
=@x
j
tomean@f
i
=@x
j
.IfSD
F
,thentheset
F.S/D
˚
U
ˇ
ˇ
UDF.X/;X2S
istheimageofSunderF.
WewilloftendenotethecomponentsofXbyx,y,...,andthecomponentsofUbyu,
v,....
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Section6.3
TheInverseFunctionTheorem
395
Example6.3.1
If
u
v
DF.x;y/D
x2Cy2
x2y2
;
then
uDf
1
.x;y/Dx
2
Cy
2
; vDf
2
.x;y/Dx
2
y
2
;
and
u
x
.x;y/D
@f
1
.x;y/
@x
D2x;
u
y
.x;y/D
@f
1
.x;y/
@y
D2y;
v
x
.x;y/D
@f
2
.x;y/
@x
D2x;
v
y
.x;y/D
@f
2
.x;y/
@y
D2y:
TofindF.R2/,weobservethat
uCvD2x
2
; uvD2y
2
;
so
F.R
2
/T D
˚
.u;v/
ˇ
ˇ
uCv0;uv0
;
whichisthepartoftheuv-planeshadedinFigure6.3.1.If.u;v/2T,then
F
p
uCv
2
;
p
uv
2
D
u
v
;
soF.R2/DT.
v
u
u + v = 0
u − v = 0
Figure6.3.1
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396 Chapter6
Vector-ValuedFunctionsofSeveralVariables
InvertibleTransformations
AtransformationFisone-to-one,orinvertible,ifF.X
1
/andF.X
2
/aredistinctwhenever
X
1
andX
2
aredistinctpointsofD
F
.Inthiscase,wecandefineafunctionGontherange
R.F/D
˚
U
ˇ
ˇ
UDF.X/forsomeX2D
F
ofFbydefiningG.U/tobetheuniquepointinD
F
suchthatF.U/DU.Then
D
G
DR.F/ and R.G/DD
F
:
Moreover,Gisone-to-one,
G.F.X//DX; X2D
F
;
and
F.G.U//DU; U2D
G
:
WesaythatGistheinverseofF,andwriteG DF
1
. TherelationbetweenFandGis
symmetric;thatis,FisalsotheinverseofG,andwewriteFDG
1
.
Example6.3.2
Thelineartransformation
u
v
DL.x;y/D
xy
xCy
(6.3.1)
maps.x;y/to.u;v/,where
uDxy;
vDxCy:
(6.3.2)
Lisone-to-oneandR.L/ D R
2
,sinceforeach.u;v/inR
2
thereisexactlyone.x;y/
suchthatL.x;y/D.u;v/. Thisissobecausethesystem(6.3.2)canbesolveduniquely
for.x;y/intermsof.u;v/:
xD
1
2
.uCv/;
yD
1
2
.uCv/:
(6.3.3)
Thus,
L
1
.u;v/D
1
2
uCv
uCv
:
Example6.3.3
Thelineartransformation
u
v
DL
1
.x;y/D
xC y
2xC2y
maps.x;y/onto.u;v/,where
uD xC y;
vD2xC2y:
(6.3.4)
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Section6.3
TheInverseFunctionTheorem
397
L
1
isnotone-to-one,sinceeverypointontheline
xCyDc (constant)
ismappedontothesinglepoint.c;2c/.Hence,L
1
doesnothaveaninverse.
ThecrucialdifferencebetweenthetransformationsofExamples6.3.2and6.3.3isthat
thematrixofLisnonsingularwhilethematrixofL
1
issingular.Thus,L(see(6.3.1))can
bewrittenas
u
v
D
1 1
1
1

x
y
;
(6.3.5)
wherethematrixhastheinverse
2
4
1
2
1
2
1
2
1
2
3
5
:
(Verify.)Multiplyingbothsidesof(6.3.5)bythismatrixyields
2
4
1
2
1
2
1
2
1
2
3
5
u
v
D
x
y
;
whichisequivalentto(6.3.3).
Sincethematrix
1 1
2 2
ofL
1
issingular,(6.3.4)cannotbesolveduniquelyfor.x;y/intermsof.u;v/.Infact,it
cannotbesolvedatallunlessvD2u.
Thefollowingtheoremsettlesthequestionofinvertibilityoflineartransformationsfrom
R
n
toR
n
.Weleavetheprooftoyou(Exercise6.3.2).
Theorem6.3.1
Thelineartransformation
UDL.X/DAX .R
n
!R
n
/
isinvertibleifandonlyifAisnonsingular;inwhichcaseR.L/DR
n
and
L
1
.U/DA
1
U:
PolarCoordinates
Wewillnowbrieflyreviewpolarcoordinates,whichwewilluseinsomeofthefollowing
examples.
Thecoordinatesofanypoint.x;y/canbewrittenininfinitelymanywaysas
xDrcos; yDrsin;
(6.3.6)
398 Chapter6
Vector-ValuedFunctionsofSeveralVariables
where
r
2
Dx
2
Cy
2
and, ifr r > 0, , istheanglefromthex-axistothelinesegmentfrom.0;0/to.x;y/,
measuredcounterclockwise(Figure6.3.2).
y
x
x2 + y2
(x,
y) 
θ
Figure6.3.2
Foreach.x;y/ ¤ .0;0/thereareinfinitelymanyvaluesof, , differingbyintegral
multiplesof2,thatsatisfy(6.3.6).Ifisanyofthesevalues,wesaythatisanargument
of.x;y/,andwrite
Darg.x;y/:
Byitself,thisdoesnotdefineafunction.However,ifisanarbitraryfixednumber,then
Darg.x;y/; <C2;
doesdefineafunction,sinceeveryhalf-openintervalŒ;C2/containsexactlyone
argumentof.x;y/.
Wedonotdefinearg.0;0/,since(6.3.6)placesnorestrictiononif.x;y/D.0;0/and
thereforerD0.
Thetransformation
r
DG.x;y/D
2
4
p
x2Cy2
arg.x;y/
3
5
; arg.x;y/<C2;
isdefinedandone-to-oneon
D
G
D
˚
.x;y/
ˇ
ˇ
.x;y/¤.0;0/
;
anditsrangeis
R.G/D
˚
.r;/
ˇ
ˇ
r>0;<C2
:
Section6.3
TheInverseFunctionTheorem
399
Forexample,ifD0,then
G.1;1/D
2
6
4
p
2
4
3
7
5
;
since=4istheuniqueargumentof.1;1/inŒ0;2/.IfD,then
G.1;1/D
2
6
4
p
2
9
4
3
7
5
;
since9=4istheuniqueargumentof.1;1/inŒ;3/.
Ifarg.x
0
;y
0
/ D D , , then.x
0
;y
0
/isonthehalf-lineshowninFigure6.3.3andGis
notcontinuousat.x
0
;y
0
/,sinceeveryneighborhoodof.x
0
;y
0
/containspoints.x;y/for
whichthesecondcomponentofG.x;y/isarbitrarilyclosetoC2,whilethesecond
componentofG.x
0
;y
0
/is. Wewillshowlater,however,thatGiscontinuous,infact,
continuouslydifferentiable,ontheplanewiththishalf-linedeleted.
y
x
(x
0
,
y
0
φ
Figure6.3.3
LocalInvertibility
AtransformationFmayfailtobeone-to-one,butbeone-to-oneonasubsetSofD
F
. By
thiswemeanthatF.X
1
/andF.X
2
/aredistinctwheneverX
1
andX
2
aredistinctpointsof
S.Inthiscase,Fisnotinvertible,butifF
S
isdefinedonSby
F
S
.X/DF.X/; X2S;
andleftundefinedforX62S,thenF
S
isinvertible. WesaythatF
S
istherestrictionofF
toS,andthatF1
S
istheinverseofFrestrictedtoS.ThedomainofF1
S
isF.S/.
400 Chapter6
Vector-ValuedFunctionsofSeveralVariables
IfFisone-to-oneonaneighborhoodofX
0
,wesaythatFislocallyinvertibleatX
0
.If
thisistrueforeveryX
0
inasetS,thenFislocallyinvertibleonS.
Example6.3.4
Thetransformation
u
v
DF.x;y/D
x
2
y
2
2xy
(6.3.7)
isnotone-to-one,since
F.x;y/DF.x;y/:
(6.3.8)
Itisone-to-oneonSifandonlyifSdoesnotcontainanypairofdistinctpointsoftheform
.x
0
;y
0
/and.x
0
;y
0
/;(6.3.8)impliesthenecessityofthiscondition,anditssufficiency
followsfromthefactthatif
F.x
1
;y
1
/DF.x
0
;y
0
/;
(6.3.9)
then
.x
1
;y
1
/D.x
0
;y
0
/ or .x
1
;y
1
/D.x
0
;y
0
/:
(6.3.10)
Toseethis,supposethat(6.3.9)holds;then
x
2
1
y
2
1
Dx
2
0
y
2
0
(6.3.11)
and
x
1
y
1
Dx
0
y
0
:
(6.3.12)
Squaringbothsidesof(6.3.11)yields
x
4
1
2x
2
1
y
2
1
Cy
4
1
Dx
4
0
2x
2
0
y
2
0
Cy
4
0
:
Thisand(6.3.12)implythat
x
4
1
x
4
0
Dy
4
0
y
4
1
:
(6.3.13)
From(6.3.11),
x
2
1
x
2
0
Dy
2
1
y
2
0
:
(6.3.14)
Factoring(6.3.13)yields
.x
2
1
x
2
0
/.x
2
1
Cx
2
0
/D.y
2
0
y
2
1
/.y
2
0
Cy
2
1
/:
Ifeithersideof(6.3.14)isnonzero,wecancanceltoobtain
x
2
1
Cx
2
0
Dy
2
0
y
2
1
;
whichimpliesthatx
0
Dx
1
Dy
0
Dy
1
D0,so(6.3.10)holdsinthiscase. Ontheother
hand,ifbothsidesof(6.3.14)arezero,then
x
1
D˙x
0
; y
1
D˙y
0
:
From(6.3.12),thesamesignmustbechosenintheseequalities,whichprovesthat(6.3.8)
implies(6.3.10)inthiscasealso.
Section6.3
TheInverseFunctionTheorem
401
Wenowsee,forexample,thatFisone-to-oneoneverysetSoftheform
SD
˚
.x;y/
ˇ
ˇ
axCby>0
;
whereaandbareconstants,notbothzero.Geometrically,Sisanopenhalf-plane;thatis,
thesetofpointsononesideof,butnoton,theline
axCbyD0
(Figure6.3.4). Therefore,FislocallyinvertibleateveryX
0
¤ .0;0/,sinceeverysuch
pointliesinahalf-planeofthisform.However,Fisnotlocallyinvertibleat.0;0/. (Why
not?)Thus,Fislocallyinvertibleontheentireplanewith.0;0/removed.
y
x
ax + by = 0
(a,
b) 
ax + by > 0
Figure6.3.4
ItisinstructivetofindF1
S
foraspecificchoiceofS. SupposethatSistheopenright
half-plane:
SD
˚
.x;y/
ˇ
ˇ
x>0
:
(6.3.15)
ThenF.S/istheentireuv-planeexceptforthenonpositiveuaxis. Toseethis,notethat
everypointinScanbewritteninpolarcoordinatesas
xDrcos; yDrsin; r>0; 
2
<<
2
:
Therefore,from(6.3.7),F.x;y/hascoordinates.u;v/,where
uDx
2
y
2
Dr
2
.cos
2
sin
2
/Dr
2
cos2;
vD2xyD2r
2
cossinDr
2
sin2:
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