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382 Chapter6
Vector-ValuedFunctionsofSeveralVariables
(ItisimportanttobearinmindthatwhileFisafunctionfromR
n
toR
m
,F
0
isnotsuch
afunction;F
0
isanmnmatrix.)FromTheorem6.2.2,thedifferentialcanbewrittenin
termsofthedifferentialmatrixas
d
X
0
FDF
0
.X
0
/
2
6
6
6
4
dx
1
dx
2
:
:
:
dx
n
3
7
7
7
5
(6.2.6)
or,moresuccinctly,as
d
X
0
FDF
0
.X
0
/dX;
where
dXD
2
6
6
6
4
dx
1
dx
2
:
:
:
dx
n
3
7
7
7
5
;
asdefinedearlier.
WhenitisnotnecessarytoemphasizetheparticularpointX
0
,wewrite(6.2.4)as
dFD
2
6
6
6
4
df
1
df
2
:
:
:
df
m
3
7
7
7
5
;
(6.2.5)as
F
0
D
2
6
6
6
6
6
6
6
6
6
6
6
4
@f
1
@x
1
@f
1
@x
2

@f
1
@x
n
@f
2
@x
1
@f
2
@x
2

@f
2
@x
n
:
:
:
:
:
:
:
:
:
:
:
:
@f
m
@x
1
@f
m
@x
2

@f
m
@x
n
3
7
7
7
7
7
7
7
7
7
7
7
5
;
and(6.2.6)as
dFDF
0
dX:
Withthedifferentialnotationwecanrewrite(6.2.2)as
lim
X!X
0
F.X/F.X
0
/F
0
.X
0
/.XX
0
/
jXX
0
j
D0:
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Section6.2
ContinuityandDifferentiabilityofTransformations
383
Example6.2.3
Thelineartransformation
F.X/D
2
6
6
6
4
a
11
x
1
Ca
12
x
2
CCa
1n
x
n
a
21
x
1
Ca
22
x
2
CCa
2n
x
n
:
:
:
a
m1
x
1
Ca
m2
x
2
CCa
mn
x
n
3
7
7
7
5
canbewrittenasF.X/DAX,whereADŒa
ij
.Then
F
0
DAI
thatis,thedifferentialmatrixofalineartransformationisindependentofXandisthe
matrixofthetransformation.Forexample,thedifferentialmatrixof
F.x
1
;x
2
;x
3
/D
1 2 3
2 1 0
2
4
x
1
x
2
x
3
3
5
is
F
0
D
1 2 3
2 1 0
:
IfF.X/DX(theidentitytransformation),thenFDI(theidentitymatrix).
Example6.2.4
Thetransformation
F.x;y/D
2
6
6
6
6
4
x
x2Cy2
y
x2Cy2
2xy
3
7
7
7
7
5
isdifferentiableateverypointofR
2
except.0;0/,and
F
0
.x;y/D
2
6
6
6
6
6
4
y2x2
.xCy2/2
2xy
.x2Cy2/2
2xy
.x2Cy2/2
x
2
y
2
.x2Cy2/2
2y
2x
3
7
7
7
7
7
5
:
Inparticular,
F
0
.1;1/D
2
6
6
6
4
0 
1
2
1
2
0
2
2
3
7
7
7
5
;
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384 Chapter6
Vector-ValuedFunctionsofSeveralVariables
so
lim
.x;y/!.1;1/
1
p
.x1/C.y1/2
0
B
B
B
@
F.x;y/
2
6
6
6
4
1
2
1
2
2
3
7
7
7
5
2
6
6
6
4
0 
1
2
1
2
0
2
2
3
7
7
7
5
x1
y1
1
C
C
C
A
D
2
4
0
0
0
3
5
:
IfmDn,thedifferentialmatrixissquareanditsdeterminantiscalledtheJacobianof
F.Thestandardnotationforthisdeterminantis
@.f
1
;f
2
;:::;f
n
/
@.x
1
;x
2
;:::;x
n
/
D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
@f
1
@x
1
@f
1
@x
2

@f
1
@x
n
@f
2
@x
1
@f
2
@x
2

@f
2
@x
n
:
:
:
:
:
:
:
:
:
:
:
:
@f
n
@x
1
@f
n
@x
2

@f
n
@x
n
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
:
WewilloftenwritetheJacobianofFmoresimplyasJ.F/,anditsvalueatX
0
asJF.X
0
/.
Sinceannnmatrixisnonsingularifandonlyifitsdeterminantisnonzero,itfollows
thatifF W R
n
! R
n
isdifferentiableatX
0
,thenF
0
.X
0
/isnonsingularifandonlyif
JF.X
0
/¤0.Wewillsoonusethisimportantfact.
Example6.2.5
If
F.x;y;´/D
2
6
6
6
4
x
2
2xC´
xC2xyC´
2
xCyC´
3
7
7
7
5
;
then
@.f
1
;f
2
;f
3
/
@.x
1
;x
2
;x
3
/
DJF.X/D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2x2
0
1
1C2y 2x 2´
1
1
1
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
D.2x2/
ˇ
ˇ
ˇ
ˇ
2x 2´
1
1
ˇ
ˇ
ˇ
ˇ
C
ˇ
ˇ
ˇ
ˇ
1C2y 2x
1
1
ˇ
ˇ
ˇ
ˇ
D.2x2/.2x2´/C.1C2y2x/:
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Section6.2
ContinuityandDifferentiabilityofTransformations
385
Inparticular,JF.1;1;1/D3,sothedifferentialmatrix
F
0
.1;1;1/D
2
4
0 0 1
1 2 2
1 1 1
3
5
isnonsingular.
PropertiesofDifferentiableTransformations
Weleavetheproofofthefollowingtheoremtoyou(Exercise6.2.16).
Theorem6.2.3
IfFWR
n
!R
m
isdifferentiableatX
0
;thenFiscontinuousatX
0
:
Theorem5.3.10andDefinition5.4.1implythefollowingtheorem.
Theorem6.2.4
LetFD.f
1
;f
2
;:::;f
m
/WR
n
!R
m
;andsupposethatthepartial
derivatives
@f
i
@x
j
; 1im; 1j j n;
(6.2.7)
existonaneighborhoodofX
0
andarecontinuousatX
0
:ThenFisdifferentiableatX
0
:
WesaythatFiscontinuouslydifferentiableonasetSifSiscontainedinanopenset
onwhichthepartialderivativesin(6.2.7)arecontinuous. Thenextthreelemmas s give
propertiesofcontinuouslydifferentiabletransformationsthatwewillneedlater.
Lemma6.2.5
SupposethatFWR
n
!R
m
iscontinuouslydifferentiableonaneigh-
borhoodN ofX
0
:Then;forevery>0;thereisaı>0suchthat
jF.X/F.Y/j<.kF
0
.X
0
/kC/jXYj if A;Y2B
ı
.X
0
/:
(6.2.8)
Proof
Considertheauxiliaryfunction
G.X/DF.X/F
0
.X
0
/X:
(6.2.9)
ThecomponentsofGare
g
i
.X/Df
i
.X/
Xn
jD1
@f
i
.X
0
/@x
j
x
j
;
so
@g
i
.X/
@x
j
D
@f
i
.X/
@x
j
@f
i
.X
0
/
@x
j
:
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386 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Thus,@g
i
=@x
j
iscontinuousonNandzeroatX
0
.Therefore,thereisaı>0suchthat
ˇ
ˇ
ˇ
ˇ
@g
i
.X/
@x
j
ˇ
ˇ
ˇ
ˇ
<
p
mn
for 1i i m; 1jn; if jXX
0
j<ı: (6.2.10)
NowsupposethatX,Y2B
ı
.X
0
/.ByTheorem5.4.5,
g
i
.X/g
i
.Y/D
Xn
jD1
@g
i
.X
i
/
@x
j
.x
j
y
j
/;
(6.2.11)
whereX
i
isonthelinesegmentfromXtoY,soX
i
2B
ı
.X
0
/. From(6.2.10),(6.2.11),
andSchwarz’sinequality,
.g
i
.X/g
i
.Y//
2
0
@
Xn
jD1
@g
i
.X
i
/
@x
j
2
1
A
jXYj
2
<
2
m
jXYj
2
:
SummingthisfromiD1toiDmandtakingsquarerootsyields
jG.X/G.Y/j<jXYj if X;Y2B
ı
.X
0
/:
(6.2.12)
Tocompletetheproof,wenotethat
F.X/F.Y/DG.X/G.Y/CF
0
.X
0
/.XY/;
(6.2.13)
so(6.2.12)andthetriangleinequalityimply(6.2.8).
Lemma6.2.6
SupposethatF WR
n
!R
n
iscontinuouslydifferentiableonaneigh-
borhoodofX
0
andF
0
.X
0
/isnonsingular:Let
rD
1
k.F0.X
0
//1k
:
(6.2.14)
Then;forevery>0;thereisaı>0suchthat
jF.X/F.Y/j.r/jXYj if X;Y2B
ı
.X
0
/:
(6.2.15)
Proof
LetXandYbearbitrarypointsinD
F
andletGbeasin(6.2.9).From(6.2.13),
jF.X/F.Y/j
ˇ
ˇ
jF
0
.X
0
/.XY/jjG.X/G.Y/j
ˇ
ˇ
;
(6.2.16)
Since
XYDŒF
0
.X
0
/
1
F
0
.X
0
/.XY/;
(6.2.14)impliesthat
jXYj
1
r
jF
0
.X
0
/.XYj;
so
jF
0
.X
0
/.XY/jrjXYj:
(6.2.17)
Nowchooseı>0sothat(6.2.12)holds.Then(6.2.16)and(6.2.17)imply(6.2.15).
SeeExercise6.2.19forastrongerconclusioninthecasewhereFislinear.
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Section6.2
ContinuityandDifferentiabilityofTransformations
387
Lemma6.2.7
IfFWR
n
!R
m
iscontinuouslydifferentiableonanopensetcontaining
acompactsetD;thenthereisaconstantMsuchthat
jF.Y/F.X/jMjYXj if X;Y2D:
(6.2.18)
Proof
On
SD
˚
.X;Y/
ˇ
ˇ
X;Y2D
R
2n
define
g.X;Y/D
8
<
:
jF.Y/F.X/F
0
.X/.YX/j
jYXj
; Y¤X;
0;
YDX:
Thengiscontinuousforall.X;Y/inSsuchthatX¤Y. WenowshowthatifX
0
2D,
then
lim
.X;Y/!.X
0
;X
0
/
g.X;Y/D0Dg.X
0
;X
0
/I
(6.2.19)
thatis,gisalsocontinuousatpoints.X
0
;X
0
/inS.
Supposethat > 0andX
0
2 D. Sincethepartialderivativesoff
1
,f
2
,...,f
m
are
continuousonanopensetcontainingD,thereisaı>0suchthat
ˇ
ˇ
ˇ
ˇ
@f
i
.Y/
@x
j
@f
i
.X/
@x
j
ˇ
ˇ
ˇ
ˇ
<
p
mn
if X;Y2B
ı
.X
0
/;1im;1j n: (6.2.20)
(Notethat@f
i
=@x
j
isuniformlycontinuouson
B
ı
.X
0
/forısufficientlysmall,fromThe-
orem5.2.14.)ApplyingTheorem5.4.5tof
1
,f
2
,...,f
m
,wefindthatifX,Y2B
ı
.X
0
/,
then
f
i
.Y/f
i
.X/D
Xn
jD1
@f
i
.X
i
/
@x
j
.y
j
x
j
/;
whereX
i
isonthelinesegmentfromXtoY.Fromthis,
2
4
f
i
.Y/f
i
.X/
Xn
jD1
@f
i
.X/
@x
j
.y
j
x
j
/
3
5
2
D
2
4
Xn
jD1
@f
i
.X
i
/
@x
j
@f
i
.X/
@x
j
.y
j
x
j
/
3
5
2
jYXj
2
n
X
jD1
@f
i
.X
i
/
@x
j
@f
i
.X/
@x
j
2
(bySchwarz’sinequality)
<
2
m
jYXj
2
(by(6.2.20)) :
SummingfromiD1toiDmandtakingsquarerootsyields
jF.Y/F.X/F
0
.X/.YX/j<jYXj if X;Y2B
ı
.X
0
/:
Thisimplies(6.2.19)andcompletestheproofthatgiscontinuousonS.
388 Chapter6
Vector-ValuedFunctionsofSeveralVariables
SinceD iscompact,soisS (Exercise5.1.27). Therefore, , gisboundedonS (Theo-
rem5.2.12);thus,forsomeM
1
,
jF.Y/F.X/F
0
.X/.YX/jM
1
jXYj if X;Y2D:
But
jF.Y/F.X/jjF.Y/F.X/F0.X/.YX/jCjF0.X/.YX/j
.M
1
CkF0.X/k/j.YXj:
(6.2.21)
Since
kF
0
.X/k
0
@
m
X
iD1
n
X
jD1
@f
i
.X/
@x
j
2
1
A
1=2
andthepartialderivativesf@f
i
=@x
j
gareboundedonD,itfollowsthatkF
0
.X/kisbounded
onD;thatis,thereisaconstantM
2
suchthat
kF
0
.X/kM
2
; X2D:
Now(6.2.21)implies(6.2.18)withMDM
1
CM
2
.
TheChainRule forTransformations
Byusingdifferentialmatrices,wecanwritethechainrulefortransformationsinaform
analogoustotheformofthechainruleforreal-valuedfunctionsofonevariable(Theo-
rem2.3.5).
Theorem6.2.8
SupposethatFWR
n
!R
m
isdifferentiableatX
0
;GWR
k
!R
n
is
differentiableatU
0
;andX
0
DG.U
0
/:ThenthecompositefunctionHDFıGWR
k
!
R
m
;definedby
H.U/DF.G.U//;
isdifferentiableatU
0
:Moreover;
H
0
.U
0
/DF
0
.G.U
0
//G
0
.U
0
/
(6.2.22)
and
d
U
0
HDd
X
0
Fıd
U
0
G;
(6.2.23)
whereıdenotescomposition:
Proof
ThecomponentsofHareh
1
,h
2
,...,h
m
,where
h
i
.U/Df
i
.G.U//:
ApplyingTheorem5.4.3toh
i
yields
d
U
0
h
i
D
Xn
jD1
@f
i
.X
0
/
@x
j
d
U
0
g
j
; 1i i m:
(6.2.24)
Section6.2
ContinuityandDifferentiabilityofTransformations
389
Since
d
U
0
HD
2
6
6
6
4
d
U
0
h
1
d
U
0
h
2
:
:
:
d
U
0
h
m
3
7
7
7
5
and d
U
0
GD
2
6
6
6
4
d
U
0
g
1
d
U
0
g
2
:
:
:
d
U
0
g
n
3
7
7
7
5
;
themequationsin(6.2.24)canbewritteninmatrixformas
d
U
0
HDF
0
.X
0
/d
U
0
GDF
0
.G.U
0
//d
U
0
G:
(6.2.25)
But
d
U
0
GDG
0
.U
0
/dU;
where
dUD
2
6
6
6
4
du
1
du
2
:
:
:
du
k
3
7
7
7
5
;
so(6.2.25)canberewrittenas
d
U
0
HDF
0
.G.U
0
//G
0
.U
0
/dU:
Ontheotherhand,
d
U
0
HDH
0
.U
0
/dU:
Comparingthelasttwoequationsyields(6.2.22).SinceG
0
.U
0
/isthematrixofd
U
0
Gand
F
0
.G.U
0
//DF
0
.X
0
/isthematrixofd
X
0
F,Theorem6.1.7
(c)
and(6.2.22)imply(6.2.23).
Example6.2.6
LetU
0
D.1;1/,
G.U/DG.u;v/D
2
6
6
6
4
p
u
p
u2C3v2
p
vC2
3
7
7
7
5
; F.X/DF.x;y;´/D
"
x2Cy2C2´2
x
2
y
2
#
;
and
H.U/DF.G.U//:
SinceGisdifferentiableatU
0
D.1;1/andFisdifferentiableat
X
0
DG.U
0
/D.1;2;1/;
Theorem6.2.8impliesthatHisdifferentiableat.1;1/.TofindH
0
.1;1/from(6.2.22),
wefirstfindthat
390 Chapter6
Vector-ValuedFunctionsofSeveralVariables
G
0
.U/D
2
6
6
6
6
6
6
4
1
2
p
u
0
u
p
u2C3v2
3v
p
u2C3v2
0
1
2
p
vC2
3
7
7
7
7
7
7
5
and
F
0
.X/D
2x
2y
2x 2y
0
:
Then,from(6.2.22),
H
0
.1;1/DF
0
.1;2;1/G
0
.1;1/
D
2
4 4
2 4 0
2
6
6
6
4
1
2
0
1
2
3
2
0
1
2
3
7
7
7
5
D
3 4
1
6
:
WecancheckthisbyexpressingHdirectlyintermsof.u;v/as
H.u;v/D
2
6
4
p
u
2
C
p
uC3v2
2
C2
p
vC2
2
p
u
2
p
uC3v2
2
3
7
5
D
uCu
2
C3v
2
C2vC4
uu
2
3v
2
anddifferentiatingtoobtain
H
0
.u;v/D
1C2u 6vC2
12u
6v
;
whichyields
H
0
.1;1/D
3 4
1
6
;
aswesawbefore.
6.2Exercises
1.
Showthatthefollowingdefinitionsareequivalent.
(a)
F D.f
1
;f
2
;:::;f
m
/iscontinuousatX
0
iff
1
,f
2
,...,f
m
arecontinuous
atX
0
.
Section6.2
ContinuityandDifferentiabilityofTransformations
391
(b)
FiscontinuousatX
0
ifforevery>0thereisaı>0suchthatjF.X/
F.X
0
/j<ifjXX
0
j<ıandX2D
F
.
2.
Verifythat
lim
X!X
0
F.X/F.X
0
/F
0
.X
0
/.XX
0
/
jXX
0
j
D0:
(a)
F.X/D
2
4
3xC4y
2x y
xC y
3
5
; X
0
D.x
0
;y
0
0
/
(b)
F.X/D
2
4
2x
2
CxyC1
xy
x
2
Cy
2
3
5
; X
0
D.1;1/
(c)
F.X/D
2
4
sin.xCy/
sin.yC´/
sin.xC´/
3
5
; X
0
D.=4;0;=4/
3.
SupposethatF W R
n
! R
m
andh W R
n
! Rhavethesamedomainandare
continuousatX
0
.ShowthattheproducthFD.hf
1
;hf
2
;:::;hf
m
/iscontinuousat
X
0
.
4.
SupposethatFandGaretransformationsfromR
n
toR
m
withcommondomainD.
ShowthatifFandGarecontinuousatX
0
2D,thensoareFCGandFG.
5.
SupposethatFWR!RisdefinedinaneighborhoodofX
0
andcontinuousat
X
0
,G WR!RisdefinedinaneighborhoodofU
0
andcontinuousatU
0
,and
X
0
DG.U
0
/.ProvethatthecompositefunctionHDFıGiscontinuousatU
0
.
6.
Prove:IfFWR
n
!R
m
iscontinuousonasetS,thenjFjiscontinuousonS.
7.
Prove:IfFWR
n
!R
m
iscontinuousonacompactsetS,thenjFjisboundedon
S,andtherearepointsX
0
andX
1
inSsuchthat
jF.X
0
/jjF.X/jjF.X
1
/j; X2SI
thatis,jFjattainsitsinfimumandsupremumonS.H
INT
:UseExercise6.2.6:
8.
ProvethatalineartransformationLWR
n
!R
m
iscontinuousonR
n
. Donotuse
Theorem6.2.8.
9.
LetAbeanmnmatrix.
(a)
UseExercises6.2.7and6.2.8toshowthatthequantitites
M.A/Dmax
jAXj
jXj
ˇ
ˇ
X¤0
and m.A/Dmin
jAXj
jXj
ˇ
ˇ
X¤0
exist.H
INT
:ConsiderthefunctionL.Y/DAYonSD
˚
Y
ˇ
ˇ
jYjD1
:
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