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412 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Weleave ittoyou(Exercise 6.3.16)toverifythat(6.3.37)and(6.3.38)can also be
obtainedbydifferentiating(6.3.36)directly.
Example6.3.9
If
u
v
DF.x;y/D
e
x
cosy
e
x
siny
(Example6.3.5),wecanalsodefineabranchGofF1onanysubsetToftheuv-planeon
whichabranchofarg.u;v/canbedefined,andGhastheform
x
y
DG.u;v/D
log.u
2
Cv
2
/
1=2
arg.u;v/
:
(6.3.39)
Sincethebranchesoftheargumentdifferbyintegralmultiplesof2,(6.3.39)impliesthat
ifG
1
andG
2
arebranchesofF
1
,bothdefinedonT,then
G
1
.u;v/G
2
.u;v/D
0
2k
(kDinteger):
FromTheorem6.3.3,
G
0
.u;v/D
F
0
.x;y/
1
D
e
x
cosy e
x
siny
e
x
siny
e
x
cosy
1
D
e
x
cosy
e
x
siny
e
x
siny e
x
cosy
:
Substitutingforxandyintermsofuandvfrom(6.3.39),wefindthat
@x
@u
D
@y
@v
De
x
cosyDe
2x
uD
u
u2Cv2
and
@x
@v
D
@y
@u
De
x
sinyDe
2x
vD
v
u2Cv2
:
TheInverseFunctionTheorem
Examples6.3.4and6.3.5showthatacontinuouslydifferentiablefunctionFmayfailto
haveaninverseonasetSevenifJF.X/¤0onS.However,thenexttheoremshowsthat
inthiscaseFislocallyinvertibleonS.
Theorem6.3.4(TheInverseFunctionTheorem)
LetF W W R
n
! R
n
be
continuouslydifferentiableonanopensetS;andsupposethatJF.X/¤0onS:Then;if
X
0
2S;thereisanopenneighborhoodN ofX
0
onwhichFisregular:Moreover;F.N/
isopenandGDF1
N
iscontinuouslydifferentiableonF.N/;with
G
0
.U/D
F
0
.X/
1
.whereUDF.X//; U2F.N/:
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Section6.3
TheInverseFunctionTheorem
413
Proof
Lemma6.2.6impliesthatthereisanopenneighborhoodN ofX
0
onwhichFis
one-to-one.TherestoftheconclusionsthenfollowfromapplyingTheorem6.3.3toFon
N.
Corollary6.3.5
IfFiscontinuouslydifferentiableonaneighborhoodofX
0
andJF.X
0
0;thenthereisanopenneighborhoodNofX
0
onwhichtheconclusionsofTheorem6.3.4
hold:
Proof
Bycontinuity,sinceJF
0
.X
0
/ ¤ 0,JF
0
.X/isnonzeroforallXinsomeopen
neighborhoodSofX
0
.NowapplyTheorem6.3.4.
Example6.3.10
LetX
0
D.1;2;1/and
2
4
u
v
w
3
5
DF.x;y;´/D
2
4
xCyC.´1/
2
C1
yC´C.x1/
2
1
´CxC.y2/
2
C3
3
5
:
Then
F
0
.x;y;´/D
2
4
1
1
2´2
2x2
1
1
1
2y4
1
3
5
;
so
JF.X
0
/D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
1 1 0
0 1 1
1 0 1
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
D2:
Inthiscase,itisdifficulttodescribeN orfindGDF1
N
explicitly;however,weknowthat
F.N/isaneighborhoodofU
0
DF.X
0
/D.4;2;5/,thatG.U
0
/DX
0
D.1;2;1/,and
that
G
0
.U
0
/D
F
0
.X
0
/
1
D
2
4
1 1 0
0 1 1
1 0 1
3
5
1
D
1
2
2
4
1 1
1
1
1 1
1
1
1
3
5
:
Therefore,
G.U/D
2
4
1
2
1
3
5
C
1
2
2
4
1 1
1
1
1 1
1
1
1
3
5
2
4
u4
v2
w5
3
5
CE.U/;
where
lim
U!.4;2;5/
E.U/
p
.u4/2C.v2/2C.w5/2
D0I
thuswehaveapproximatedGnearU
0
D.4;2;5/byanaffinetransformation.
Theorem6.3.4and(6.3.34)implythatthetransformation(6.3.32)islocallyinvertible
onSD
˚
.r;/
ˇ
ˇ
r>0
,whichmeansthatitispossibletodefineabranchofarg.x;y/ina
neighborhoodofanypoint.x
0
;y
0
/¤.0;0/.Italsoimplies,aswehavealreadyseen,that
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414 Chapter6
Vector-ValuedFunctionsofSeveralVariables
thetransformation(6.3.7)ofExample6.3.4islocallyinvertibleeverywhereexceptat.0;0/,
whereitsJacobianequalszero,andthetransformation(6.3.16)ofExample6.3.5islocally
invertibleeverywhere.
6.3Exercises
1.
Prove:IfFisinvertible,thenF1isunique.
2.
ProveTheorem6.3.1.
3.
Prove:ThelineartransformationL.X/DAXcannotbeone-to-oneonanyopenset
ifAissingular.H
INT
:UseTheorem6.1.15:
4.
Let
G.x;y/D
" p
x2Cy2
arg.x;y/
#
; =2arg.x;y/<5=2:
Find
(a)
G.0;1/
(b)
G.1;0/
(c)
G.1;0/
(d)
G.2;2/
(e)
G.1;1/
5.
SameasExercise6.3.4,exceptthat2arg.x;y/<0.
6. (a)
Prove:Iff WR!Riscontinuousandlocallyinvertibleon.a;b/,thenf f is
invertibleon.a;b/.
(b)
Giveanexampleshowingthatthecontinuityassumptionisneededin
(a)
.
7.
Let
F.x;y/D
x
2
y
2
2xy
(Example6.3.4)and
S D
˚
.x;y/
ˇ
ˇ
axCby>0
.a
2
Cb
2
¤0/:
FindF.S/andF
1
S
.If
S
1
D
˚
.x;y/
ˇ
ˇ
axCby<0
;
showthatF.S
1
/DF.S/andF
1
S
1
DF
1
S
.
8.
Showthatthetransformation
u
v
DF.x;y/D
e
x
cosy
e
x
siny
(Example6.3.5)isone-to-oneonanysetSthatdoesnotcontainanypairofpoints
.x
0
;y
0
/and.x
0
;y
0
C2k/,wherekisanonzerointeger.
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Section6.3
TheInverseFunctionTheorem
415
9.
SupposethatFWR
n
!R
n
iscontinuousandinvertibleonacompactsetS.Show
thatF
1
S
iscontinuous. H
INT
:IfF
1
S
isnotcontinuousat
UinF.S/;thenthereis
an
0
>0andasequencefU
k
ginF.S/suchthatlim
k!1
U
k
D
Uwhile
jF
1
S
.U
k
/F
1
S
.
U/j
0
; k1:
UseExercise5.1.32toobtainacontradiction:
10.
FindF
1
and.F
1
/
0
:
(a)
u
v
DF.x;y/D
4xC2y
3xC y
(b)
2
4
u
v
w
3
5
DF.x;y;´/D
2
4
xCyC2´
3xCy4´
xyC2´
3
5
11.
InadditiontotheassumptionsofTheorem6.3.3,supposethatallqth-order.q>1/
partialderivativesofthecomponentsofFarecontinuousonS. Showthatallqth-
orderpartialderivativesofF
1
S
arecontinuousonF.S/.
12.
If
u
v
DF.x;y/D
x
2
Cy
2
x
2
y
2
(Example6.3.1),findfourbranchesG
1
,G
2
,G
3
,andG
4
ofF
1
definedon
T
1
D
˚
.u;v/
ˇ
ˇ
uCv>0;uv>0
;
andverifythatG0
i
.u;v/D.F0.x.u;v/;y.u;v///1,1i4.
13.
SupposethatAisanonsingularnnmatrixand
UDF.X/DA
2
6
6
6
4
x
2
1
x
2
2
:
:
:
x
2
n
3
7
7
7
5
:
(a)
ShowthatFisregularontheset
SD
˚
X
ˇ
ˇ
e
i
x
i
>0;1in
;
wheree
i
D˙1,1in.
(b)
FindF1
S
.U/.
(c)
Find.F1
S
/0.U/.
14.
Let.x;y/beabranchofarg.x;y/definedonanopensetS.
(a)
Showthat.x;y/cannotassumealocalextremevalueatanypointofS.
(b)
Prove:Ifa¤0andthelinesegmentfrom.x
0
;y
0
/to.ax
0
;ay
0
/isinS,then
.ax
0
;ay
0
/D.x
0
;y
0
/.
(c)
ShowthatScannotcontainasubsetoftheform
AD
n
.x;y/
ˇ
ˇ
0<r
1
p
x2Cyr
2
o
:
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416 Chapter6
Vector-ValuedFunctionsofSeveralVariables
(d)
Showthatnobranchofarg.x;y/canbedefinedonadeletedneighborhood
oftheorigin.
15.
ObtainEqn.(6.3.35)formallybydifferentiating:
(a)
arg.x;y/Dcos
1
x
p
x2Cy2
(b)
arg.x;y/Dsin
1
y
p
x2Cy2
(c)
arg.x;y/Dtan
1
y
x
Wheredotheseformulascomefrom?Whatisthedisadvantageofusinganyoneof
themtodefinearg.x;y/?
16.
Forthetransformation
u
v
DF.x;y/D
x2y2
2xy
(Example6.3.4),findabranchGofF
1
definedonT D
˚
.u;v/
ˇ
ˇ
auCbv>0
.
FindG
0
bymeansoftheformulaG
0
.U/DŒF
0
.X/
1
ofTheorem6.3.3,andalsoby
directdifferentiationwithrespecttouandv.
17.
Atransformation
F.x;y/D
u.x;y/
v.x;y/
isanalyticonasetSifitiscontinuouslydifferentiableand
u
x
Dv
y
; u
y
Dv
x
onS.Prove:IfFisanalyticandregularonS,thenF
1
S
isanalyticonF.S/;thatis,
x
u
Du
v
andx
v
Du
u
.
18.
Prove:IfUDF.X/andXDG.U/areinversefunctions,then
@.u
1
;u
2
;:::;u
n
/
@.x
1
;x
2
;:::;x
n
@.x
1
;x
2
;:::;x
n
/
@.u
1
;u
2
;:::;u
n
/
D1:
WhereshouldtheJacobiansbeevaluated?
19.
GiveanexampleofatransformationFWR
n
!R
n
thatisinvertiblebutnotregular
onR
n
.
20.
Findanaffine transformationAthatsowellapproximatesthebranchGofF
1
definednearU
0
DF.X
0
/that
lim
U!U
0
G.U/A.U/
jUU
0
j
D0:
(a)
u
v
DF.x;y/D
x
4
y
5
4x
x
3
y
2
3y
; X
0
D.1;1/
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Section6.4
TheImplicitFunctionTheorem
417
(b)
u
v
DF.x;y/D
x2yCxy
2xyCxy2
; X
0
D.1;1/
(c)
2
4
u
v
w
3
5
DF.x;y;´/D
2
4
2x
2
yCx
3
x
3
Cy´
xCyC´
3
5
; XD.0;1;1/
(d)
2
4
u
v
w
3
5
DF.x;y;´/D
2
4
xcosycos´
xsinycos´
xsin´
3
5
; X
0
D.1;=2;/
21.
IfFisdefinedby
2
4
x
y
´
3
5
DF.r;;/D
2
4
rcoscos
rsincos
rsin
3
5
andG isa branch h ofF
1
, find d G
0
interms ofr, , and d . H
INT
: See Exer-
cise6.2.14.b/:
22.
IfFisdefinedby
2
4
x
y
´
3
5
DF.r;;´/D
2
4
rcos
rsin
´
3
5
andG isa branch h ofF
1
, findG
0
interms ofr, , , and ´. H
INT
: See Exer-
cise6.2.14.c/:
23.
SupposethatF WR
n
! R
n
isregularonacompactsetT. ShowthatF.@T/ / D
@F.T/;thatis,boundarypointsmaptoboundarypoints.H
INT
:UseExercise6.2.23
andTheorem6.3.3toshowthat@F.T/F.@T/:ThenapplythisresultwithFand
T replacedbyF
1
andF.T/toshowthatF.@T/@F.T/:
6.4THEIMPLICITFUNCTIONTHEOREM
InthissectionweconsidertransformationsfromR
nCm
toR
m
. Itwillbeconvenientto
denotepointsinR
nCm
by
.X;U/D.x
1
;x
2
;:::;x
n
;u
1
;u
2
;:::;u
m
/:
WewilloftendenotethecomponentsofXbyx,y,...,andthecomponentsofUbyu,v,
....
Tomotivatetheproblemweareinterestedin,wefirstaskwhetherthelinearsystemof
mequationsinmCnvariables
a
11
x
1
Ca
12
x
2
CCa
1n
x
n
Cb
11
u
1
Cb
12
u
2
CCb
1m
u
m
D0
a
21
x
1
Ca
22
x
2
CCa
2n
x
n
Cb
21
u
1
Cb
22
u
x
CCb
2m
u
m
D0
:
:
:
a
m1
x
1
Ca
m2
x
2
CCa
mn
x
n
Cb
m1
u
1
Cb
m2
u
2
CCb
mm
u
m
D0
(6.4.1)
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418 Chapter6
Vector-ValuedFunctionsofSeveralVariables
determinesu
1
,u
2
,...,u
m
uniquelyintermsofx
1
,x
2
,...,x
n
.Byrewritingthesystemin
matrixformas
AXCBUD0;
where
AD
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
:
:
:
:
:
:
:
:
:
:
:
:
a
m1
a
m2
 a
mn
3
7
7
7
5
; BD
2
6
6
6
4
b
11
b
12
 b
1m
b
21
b
22
 b
2m
:
:
:
:
:
:
:
:
:
:
:
:
b
m1
b
m2
 b
mm
3
7
7
7
5
;
XD
2
6
6
6
4
x
1
x
2
:
:
:
x
n
3
7
7
7
5
; and UD
2
6
6
6
4
u
1
u
2
:
:
:
u
m
3
7
7
7
5
;
weseethat(6.4.1)canbesolveduniquelyforUintermsofXifthesquarematrixBis
nonsingular.Inthiscasethesolutionis
UDB
1
AX:
Forourpurposesitisconvenienttorestatethis:If
F.X;U/DAXCBU;
(6.4.2)
whereBisnonsingular,thenthesystem
F.X;U/D0
determinesUasafunctionofX,forallXinR
n
.
NoticethatFin(6.4.2)isalineartransformation.IfFisamoregeneraltransformation
fromR
nCm
toR
m
,wecanstillaskwhetherthesystem
F.X;U/D0;
or,intermsofcomponents,
f
1
.x
1
;x
2
;:::;x
n
;u
1
;u
2
;:::;u
m
/D0
f
2
.x
1
;x
2
;:::;x
n
;u
1
;u
2
;:::;u
m
/D0
:
:
:
f
m
.x
1
;x
2
;:::;x
n
;u
1
;u
2
;:::;u
m
/D0;
canbesolvedforUintermsofX.However,thesituationisnowmorecomplicated,even
ifmD1.Forexample,supposethatmD1and
f.x;y;u/D1x
2
y
2
u
2
:
Section6.4
TheImplicitFunctionTheorem
419
Ifx
2
Cy
2
>1,thennovalueofusatisfies
f.x;y;u/D0:
(6.4.3)
However,infinitelymanyfunctionsuDu.x;y/satisfy(6.4.3)ontheset
SD
˚
.x;y/
ˇ
ˇ
x
2
Cy
2
1
:
Theyareoftheform
u.x;y/D.x;y/
p
1x2y2;
where.x;y/canbechosenarbitrarily,foreach.x;y/inS,tobe1or1.Wecannarrow
thechoiceoffunctionstotwobyrequiringthatubecontinuousonS;then
u.x;y/D
p
1xy2
(6.4.4)
or
u.x;y/D
p
1x2y2:
Wecandefineauniquecontinuoussolutionuof(6.4.3)byspecifyingitsvalueatasingle
interiorpointofS.Forexample,ifwerequirethat
u
1
p
3
;
1
p
3
D
1
p
3
;
thenumustbeasdefinedby(6.4.4).
Thequestionofwhetheranarbitrarysystem
F.X;U/D0
determinesUasafunctionofXistoogeneraltohaveausefulanswer. However,there
isatheorem, theimplicitfunctiontheorem, thatanswers thisquestionaffirmativelyin
animportantspecialcase. Tofacilitatethestatementofthistheorem, , wepartitionthe
differentialmatrixofFWR
nCm
!R
m
:
F
0
D
2
6
6
6
6
6
6
6
6
6
4
@f
1
@x
1
@f
1
@x
2

@f
1
@x
n
j
@f
1
@u
1
@f
1
@u
2

@f
1
@u
m
@f
2
@x
1
@f
2
@x
2

@f
2
@x
n
j
@f
2
@u
1
@f
2
@u
2

@f
2
@u
m
:
:
:
:
:
:
:
:
:
:
:
:
j
:
:
:
:
:
:
:
:
:
:
:
:
@f
m
@x
1
@f
m
@x
2

@f
m
@x
n
j
@f
m
@u
1
@f
m
@u
2

@f
m
@u
m
3
7
7
7
7
7
7
7
7
7
5
(6.4.5)
or
F
0
DŒF
X
;F
U
;
whereF
X
isthesubmatrixtotheleftofthedashedlinein(6.4.5)andF
U
istotheright.
Forthelineartransformation(6.4.2),F
X
DAandF
U
DB,andwehaveseenthatthe
systemF.X;U/D0definesUasafunctionofXforallXinR
n
ifF
U
isnonsingular.The
nexttheoremshowsthatarelatedresultholdsformoregeneraltransformations.
420 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Theorem6.4.1(TheImplicitFunctionTheorem)
SupposethatFWR
nCm
!
R
m
iscontinuouslydifferentiableonanopensetS ofR
nCm
containing.X
0
;U
0
/:Let
F.X
0
;U
0
/D0;andsupposethatF
U
.X
0
;U
0
/isnonsingular:Thenthereisaneighborhood
Mof.X
0
;U
0
/;containedinS;onwhichF
U
.X;U/isnonsingularandaneighborhoodN
ofX
0
inR
n
onwhichauniquecontinuouslydifferentiabletransformationGWR
n
!R
m
isdefined;suchthatG.X
0
/DU
0
and
.X;G.X//2M
and F.X;G.X//D0 if X2N:
(6.4.6)
Moreover;
G
0
.X/DŒF
U
.X;G.X//
1
F
X
.X;G.X//; X2N:
(6.4.7)
Proof
DefineˆWR
nCm
!R
nCm
by
ˆ.X;U/D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
x
1
x
2
:
:
:
x
n
f
1
.X;U/
f
2
.X;U/
:
:
:
f
m
.X;U/
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(6.4.8)
or,in“horizontal”notationby
ˆ.X;U/D.X;F.X;U//:
(6.4.9)
ThenˆiscontinuouslydifferentiableonSand,sinceF.X
0
;U
0
/D0,
ˆ.X
0
;U
0
/D.X
0
;0/:
(6.4.10)
Thedifferentialmatrixofˆis
ˆ
0
D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1
0

0
0
0

0
0
1

0
0
0

0
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
0
0

1
0
0

0
@f
1
@x
1
@f
1
@x
2

@f
1
@x
n
@f
1
@u
1
@f
1
@u
2

@f
1
@u
m
@f
2
@x
1
@f
2
@x
2

@f
2
@x
n
@f
2
@u
1
@f
2
@u
2

@f
2
@u
m
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
@f
m
@x
1
@f
m
@x
2

@f
m
@x
n
@f
m
@u
1
@f
m
@u
2

@f
m
@u
m
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
D
I
0
F
X
F
U
;
Section6.4
TheImplicitFunctionTheorem
421
whereIisthennidentitymatrix,0isthenmmatrixwithallzeroentries,andF
X
andF
U
areasin(6.4.5).Byexpandingdet.ˆ
0
/andthedeterminantsthatevolvefromitin
termsofthecofactorsoftheirfirstrows,itcanbeshowninnstepsthat
JˆDdet.ˆ
0
/D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
@f
1
@u
1
@f
1
@u
2

@f
1
@u
m
@f
2
@u
1
@f
2
@u
2

@f
2
@u
m
:
:
:
:
:
:
:
:
:
:
:
:
@f
m
@u
1
@f
m
@u
2

@f
m
@u
m
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
Ddet.F
U
/:
Inparticular,
Jˆ.X
0
;U
0
/Ddet.F
U
.X
0
;U
0
/¤0:
SinceˆiscontinuouslydifferentiableonS,Corollary6.3.5impliesthatˆisregularon
someopenneighborhoodMof.X
0
;U
0
/andthat
c
MDˆ.M/isopen.
Becauseoftheformofˆ(see(6.4.8)or(6.4.9)),wecanwritepointsof
c
M as.X;V/,
whereV2 R
m
. Corollary6.3.5alsoimpliesthatˆhasaacontinuouslydifferentiable
inverse.X;V/definedon
c
M withvaluesinM. Sinceˆleavesthe“Xpart"of.X;U/
fixed,alocalinverseofˆmustalsohavethisproperty.Therefore,musthavetheform
.X;V/D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
x
1
x
2
:
:
:
x
n
h
1
.X;V/
h
2
.X;V/
:
:
:
h
m
.X;V/
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
or,in“horizontal”notation,
.X;V/D.X;H.X;V//;
whereHWR
nCm
!R
m
iscontinuouslydifferentiableon
c
M.WewillshowthatG.X/D
H.X;0/hasthestatedproperties.
From(6.4.10),.X
0
;0/2
c
M and,since
c
Misopen,thereisaneighborhoodN ofX
0
in
R
n
suchthat.X;0/2
c
MifX2N(Exercise6.4.2).Therefore,.X;G.X//D.X;0/2M
ifX2N.SinceDˆ
1
,.X;0/Dˆ.X;G.X//.SettingXDX
0
andrecalling(6.4.10)
showsthatG.X
0
/DU
0
,sinceˆisone-to-oneonM.
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