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512 Chapter7
IntegralsofFunctionsofSeveralVariables
SincethepairwiseintersectionsofT
1
, T
2
,T
3
,andT
4
allhavezerocontent,I D D I
1
C
I
2
CI
3
CI
4
(Corollary7.1.31). Theorem7.3.8impliesthatI
1
DI
2
DI
3
DI
4
(Exer-
cise7.3.12),soID4I
1
.SinceI
1
doesnotcontainanypairsofdistinctpointsoftheform
.x
0
;y
0
/and.x
0
;y
0
/,Fisone-to-oneonT
1
(Example6.3.4),
F.T
1
/DS
1
D
˚
.u;v/
ˇ
ˇ
a
4
u
2
Cv
2
b
4
;v0
(Figure7.3.10
(b)
),
S
1
s
1
v
u
ρ
α
π
a2
b2
a2
b2
(a)
(b)
Figure7.3.10
andabranchGofF
1
canbedefinedonS
1
(Example6.3.8).NowTheorem7.3.15implies
that
I
1
D
Z
S
1
e
.x
2
y
2
/
2
e
4x
2
y
2
.x
2
Cy
2
/jJG.u;v/jd.u;v/;
wherexandymuststillbewrittenintermsofuandv.Sinceitiseasytoverifythat
JF.x;y/D4.x
2
Cy
2
/
andtherefore
JG.u;v/D
1
4.x2Cy2/
;
doingthisyields
I
1
D
1
4
Z
S
1
e
u
2
Cv
2
d.u;v/:
(7.3.60)
Toevaluatethisintegral,weletand˛bepolarcoordinatesintheuv-plane(Figure7.3.11)
anddefineHby
u
v
D H.;˛/D
cos˛
sin˛
I
thenS
1
DH.
e
S
1
/,where
e
S
1
D
˚
.;˛/
ˇ
ˇ
a
2
b
2
;0˛
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Section7.3
ChangeofVariablesinMultipleIntegrals
513
(Figure7.3.10
(a)
);hence,applyingTheorem7.3.15to(7.3.60)yields
I
1
D
1
4
Z
e
S
1
e
2
jJH.;˛/jd.;˛/D
1
4
Z
e
S
1
e
2
d.;˛/
D
1
4
Z
0
Z
b
2
a
2
e
2
dD
.e
b
4
e
a
4
/
8
I
hence,
ID4I
1
D
2
.e
b
4
e
a
4
/:
v
u
ρ
α
(u,
v) 
Figure7.3.11
Example7.3.8
Evaluate
I D
Z
T
e
x
1
Cx
2
CCx
n
d.x
1
;x
2
;:::;x
n
/;
whereTistheregiondefinedby
a
i
x
1
Cx
2
CCx
i
b
i
; 1in:
Solution
Wedefinethenewvariablesy
1
,y
2
,...,y
n
byYDF.X/,where
f
i
.X/Dx
1
Cx
2
CCx
i
; 1i i n:
IfGDF
1
thenT DG.S/,where
SDŒa
1
;b
1
Œa
2
;b
2
Œa
n
;b
n
;
andJG.Y/D1,sinceJF.X/D1(verify);hence,Theorem7.3.8impliesthat
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514 Chapter7
IntegralsofFunctionsofSeveralVariables
I D
Z
S
e
y
n
d.y
1
;y
2
;:::;y
n
/
D
Z
b
1
a
1
dy
1
Z
b
2
a
2
dy
2

Z
b
n1
a
n1
dy
n1
Z
b
n
a
n
e
y
n
dy
n
D.b
1
a
1
/.b
2
a
2
/.b
n1
a
n1
/.e
b
n
e
a
n
/:
7.3Exercises
1.
Giveacounterexampletothefollowingstatement:IfS
1
andS
2
aredisjointsubsets
ofarectangleR,theneither
Z
R
S
1
.X/dXC
Z
R
S
2
.X/dXD
Z
R
S
1
[S
2
.X/dX
or
Z
R
S
1
.X/dXC
Z
R
S
2
.X/dXD
Z
R
S
1
[S
2
.X/dX:
2.
ShowthatasetEhascontentzeroaccordingtoDefinition7.1.14ifandonlyifE
hasJordancontentzero.
3.
ShowthatifS
1
andS
2
areJordanmeasurable,thensoareS
1
[S
2
andS
1
\S
2
.
4.
Prove:
(a)
IfSisJordanmeasurablethensois
S,andV.
S/DV.S/.MustSbeJordan
measurableif
Sis?
(b)
IfTisaJordanmeasurablesubsetofaJordanmeasurablesetS,thenST
isJordanmeasurable.
5.
SupposethatHisasubsetofacompactJordanmeasurablesetSsuchthattheinter-
sectionofHwithanycompactsubsetofS
0
haszerocontent.ShowthatV.H/D0.
6.
SupposethatEisannnelementarymatrixandAisanarbitrarynpmatrix.
ShowthatEAisthematrixobtainedbyapplyingtoAtheoperationbywhichEis
obtainedfromthennidentitymatrix.
7. (a)
Calculatethedeterminantsofelementarymatricesoftypes
(a)
,
(b)
,and
(c)
ofLemma7.3.6.
(b)
Showthattheinverseofanelementarymatrixoftype
(a)
,
(b)
,or
(c)
isan
elementarymatrixofthesametype.
(c)
Verifytheinversesgivenfor
b
E
1
;:::;
b
E
6
inExample7.3.1.
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Section7.3
ChangeofVariablesinMultipleIntegrals
515
8.
Writeasaproductofelementarymatrices.
(a)
2
4
1 0 1
1 1 0
0 1 1
3
5
(b)
2
4
2
3 2
0 1
5
0 2
4
3
5
9.
Supposethatadbc¤0,u
1
<u
2
,andv
1
<v
2
.Findtheareaoftheparallelogram
boundedbythelines
axCbyDu
1
;
axCbyDu
2
;
cxCdyDv
1
;
cxCdyDv
2
:
10.
Findthevolumeoftheparallelepipeddefinedby
12xC3y2´2; 5xC5y7; 12xC4y6:
11.
InwritingEqn.(7.3.53)weassumedthat
Z
G.S/
f.X/dXD
Z
G.S
1
/
f.X/dXC
Z
G.S\S
c
1
/
f.X/dX:
Justifythis.H
INT
:ShowthatG.S
1
/\G.S\S
c
1
/haszerocontent:
12.
UseTheorem7.3.8toshowthatI
1
DI
2
DI
3
DI
4
inExample7.3.7.
13.
Lete
i
D˙1,0in.LetTbeaboundedsubsetofR
n
and
b
T D
˚
.e
1
x
1
;e
2
x
2
;:::;e
n
x
n
/
ˇ
ˇ
.x
1
;x
2
;:::;x
n
/2T
:
Supposethatf isdefinedonTanddefinegon
b
T by
g.e
1
x
1
;e
2
x
2
;:::;e
n
x
n
/De
0
f.x
1
;x
2
;:::;x
n
/:
(a)
ProvedirectlyfromDefinitions7.1.2and7.1.17thatf isintegrableonT T if
andonlyifgisintegrableon
b
T,andinthiscase
Z
b
T
g.Y/dYDe
0
Z
T
f.X/dX:
(b)
Supposethat
b
T DT,
f.e
1
x
1
;e
2
x
2
;:::;e
n
x
n
/Df.x
1
;x
2
;:::;x
n
/;
andf isintegrableonT.Showthat
Z
T
f.X/dXD0:
14.
Findtheareaof
(a)
˚
.x;y/
ˇ
ˇ
yx4y;1xC2y3
;
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516 Chapter7
IntegralsofFunctionsofSeveralVariables
(b)
˚
.x;y/
ˇ
ˇ
2xy4;2xy5x
.
15.
Evaluate
Z
T
.3x
2
C2yC´/d.x;y;´/;
where
TD
˚
.x;y;´/
ˇ
ˇ
jxyj1;jy´j1;j´Cxj1
:
16.
Evaluate
Z
T
.y
2
Cx
2
y2x
4
/d.x;y/;
whereTistheregionboundedbythecurves
xyD1; xyD2; yDx
2
; yDx
2
C1:
17.
Evaluate
Z
T
.x
4
y
4
/e
xy
d.x;y/;
whereTistheregioninthefirstquadrantboundedbythehyperbolas
xyD1; xyD2; x
2
y
2
D2; x
2
y
2
D3:
18.
Findthevolumeoftheellipsoid
x
2
a2
C
y
2
b2
C
´
2
c2
D1 .a;b;c>0/:
19.
Evaluate
Z
T
ex
2
Cy
2
2
p
x2Cy2
d.x;y;´/;
where
T D
˚
.x;y;´/
ˇ
ˇ
9x
2
Cy
2
2
25
:
20.
FindthevolumeofthesetT boundedbythesurfaces´D0,´D
p
x2Cy2,and
x
2
Cy
2
D4.
21.
Evaluate
Z
T
xy´.x
4
y
4
/d.x;y;´/;
where
T D
˚
.x;y;´/
ˇ
ˇ
1x
2
y
2
2;3x
2
Cy
2
4;0´1
:
22.
Evaluate
(a)
Z
p
2
0
dy
Z
p
4y
2
y
dx
1Cx2Cy2
(b)
Z
2
0
dx
Z
p
4x
2
0
e
x
2
Cy
2
dy
(c)
Z
1
1
dx
Z
p
1x
2
p
1x
2
dy
Z
p
1x
2
y
2
0
´
2
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Section7.3
ChangeofVariablesinMultipleIntegrals
517
23.
Usethechangeofvariables
2
6
6
4
x
1
x
2
x
3
x
4
3
7
7
5
DG.r;
1
;
2
;
3
/D
2
6
6
4
rcos
1
cos
2
cos
3
rsin
1
cos
2
cos
3
rsin
2
cos
3
rsin
3
3
7
7
5
tocomputethecontentofthe4-ball
T D
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
x
2
1
Cx
2
2
Cx
2
3
Cx
2
4
a
2
:
24.
SupposethatA D D Œa
ij
isanonsingularnnmatrixandT istheregioninR
n
definedby
˛
1
a
i1
x
1
Ca
i2
x
2
CCa
in
x
n
ˇ
i
; 1in:
(a)
FindV.T/.
(b)
Showthatifc
1
,c
2
,...,c
n
areconstants,then
Z
T
0
@
Xn
jD1
c
j
x
j
1
A
dXD
V.T/
2
Xn
iD1
d
i
i
i
/;
where
2
6
6
6
4
d
1
d
2
:
:
:
d
n
3
7
7
7
5
D.A
t
/
1
2
6
6
6
4
c
1
c
2
:
:
:
c
n
3
7
7
7
5
:
25.
IfV
n
isthecontentofthen-ballT D
˚
X
ˇ
ˇ
jXj1
, findthecontentofthen-
dimensionalellipsoiddefinedby
Xn
jD1
x
2
j
a
2
j
1:
LeavetheanswerintermsofV
n
.
CHAPTER8
MetricSpaces
INTHISCHAPTERwestudymetricspaces.
SECTION8.1definestheconceptandbasicpropertiesofametricspace.Severalexamples
ofmetricspacesareconsidered.
SECTION8.2definesanddiscussescompactnessinametricspace.
SECTION8.3dealswithcontinuousfunctionsonmetricspaces.
8.1INTRODUCTIONTOMETRICSPACES
Definition8.1.1
AmetricspaceisanonemptysetAtogetherwithareal-valuedfunc-
tiondefinedonAAsuchthatifu,v,andwarearbitrarymembersofA,then
(a)
.u;v/0,withequalityifandonlyifuDv;
(b)
.u;v/D.v;u/;
(c)
.u;v/.u;w/C.w;v/.
WesaythatisametriconA.
Ifn2andu
1
,u
2
,...,u
n
arearbitrarymembersofA,then
(c)
andinductionyield
theinequality
.u
1
;u
n
/
n1
iD1
.u
i
;u
iC1
/:
Example8.1.1
ThesetRofrealnumberswith.u;v/Djuvjisametricspace.
Definition8.1.1
(c)
isthefamiliartriangleinequality:
juvjjuwjCjwuj:
Motivatedbythisexample,inanarbitrarymetricspacewecall.u;v/thedistancefrom
utov,andwecallDefinition8.1.1
(c)
thetriangleinequality.
518
Section8.1
IntroductiontoMetricSpaces
519
Example8.1.2
IfAisanarbitrarynonemptyset,then
.u;v/D
0 ifuDv;
1 ifu¤v
isametriconA(Exercise8.1.5).Wecallitthediscretemetric.
Example8.1.2showsthatitispossibletodefineametriconanynonemptysetA. In
fact,itispossibletodefineinfinitelymanymetricsonanysetwithmorethanonemember
(Exercise8.1.3). Therefore, , tospecifyametricspacecompletely,wemustspecifythe
couple.A;/,whereAisthesetandisthemetric. (Insomecases s wewillnotbeso
precise;forexample,wewillalwaysrefertotherealnumberswiththemetric.u;v/D
juvjsimplyasR.)
Thereisanimportantkindofmetricspace thatariseswhenadefinitionoflengthis
imposedonavectorspace. Althoughweassumethatyouarefamiliarwiththedefinition
ofavectorspace,werestateithereforconvenience. Weconfinethedefinitiontovector
spacesovertherealnumbers.
Definition8.1.2
AvectorspaceAisanonemptysetofelements calledvectorson
whichtwooperations, vectoradditionandscalarmultiplication(multiplicationbyreal
numbers)aredefined,suchthatthefollowingassertionsaretrueforallU,V,andWin
Aandallrealnumbersrands:
1.UCV2A;
2.UCVDVCU;
3.UC.VCW/D.UCV/CW;
4.Thereisavector0inAsuchthatUC0DU;
5.ThereisavectorUinAsuchthatUC.U/D0;
6.rU2A;
7.r.UCV/DrUCrV;
8..rCs/UDrUCsU;
9.r.sU/D.rs/U;
10.1UDU.
WesaythatAisclosedundervectoradditionif(1)istrue,andthatAisclosedunder
scalarmultiplicationif(6)istrue.ItcanbeshownthatifBisanynonemptysubsetofA
thatisclosedundervectoradditionandscalarmultiplication,thenBtogetherwiththese
operationsisitselfavectorspace. (Seeanylinearalgebratextfortheproof.)Wesaythat
BisasubspaceofA.
Definition8.1.3
AnormedvectorspaceisavectorspaceAtogetherwithareal-valued
functionN definedonA, suchthatifuandvarearbitraryvectorsinAandaisareal
number,then
(a)
N.u/0withequalityifandonlyifuD0;
(b)
N.au/DjajN.u/;
(c)
N.uCv/N.u/CN.v/.
WesaythatNisanormonA,and.A;N/isanormedvectorspace.
520 Chapter8
MetricSpaces
Theorem8.1.4
If.A;N/isanormedvectorspace;then
.x;y/DN.xy/
(8.1.1)
isametriconA:
Proof
From
(a)
withuDxy,.x;y/DN.xy/0,withequalityifandonly
ifxDy.From
(b)
withuDxyandaD1,
.y;x/DN.yx/DN..xy//DN.xy/D.x;y/:
From
(c)
withuDx´andvD´y,
.x;y/DN.xy/N.x´/CN.´y/D.x;´/C.´;y/:
Wewillsaythatthemetricin(8.1.1)isinducedbythenormN. Wheneverwespeakof
anormedvectorspace.A;N/,itistobeunderstoodthatweareregardingitasametric
space.A;/,whereisthemetricinducedbyN.
WewilloftenwriteN.u/askuk.Inthiscasewewilldenotethenormedvectorspaceas
.A;kk/.
Theorem8.1.5
Ifxandyarevectorsinanormedvectorspace.A;N/;then
jN.x/N.y/jN.xy/:
(8.1.2)
Proof
Since
xDyC.xy/;
Definition8.1.3
(c)
withuDyandvDxyimpliesthat
N.x/N.y/CN.xy/;
or
N.x/N.y/N.xy/:
Interchangingxandyyields
N.y/N.x/N.yx/:
SinceN.xy/DN.yx/(Definition8.1.3
(b)
withuDxyandaD1),thelast
twoinequalitiesimply(8.1.2).
Metricsfor
RRR
n
InSection5.1wedefinedthenormofavectorXD.x
1
;x
2
;:::;x
n
/inR
n
as
kXkD
Xn
iD1
x
2
i
!
1=2
:
Section8.1
IntroductiontoMetricSpaces
521
Themetricinducedbythisnormis
.X;Y/D
Xn
iD1
.x
i
y
i
/
2
!
1=2
:
WheneverwewriteR
n
withoutidentifyingthenormormetricspecifically,wearereferring
toR
n
withthisnormandthisinducedmetric.
ThefollowingdefinitionprovidesinfinitelymanynormsandmetricsonR
n
.
Definition8.1.6
Ifp1andXD.x
1
;x
2
;:::;x
n
/,let
kXk
p
D
Xn
iD1
jx
i
j
p
!
1=p
:
(8.1.3)
ThemetricinducedonRbythisnormis
p
.X;Y/D
Xn
iD1
jx
i
y
i
j
p
!
1=p
:
Tojustifythisdefinition,wemustverifythat(8.1.3)actuallydefinesanorm.Sinceitis
clearthatkXk
p
0withequalityifandonlyifXD0,andkaXk
p
DjajkXk
p
ifaisany
realnumberandX2R
n
,thisreducestoshowingthat
kXCYk
p
kXk
p
CkYk
p
(8.1.4)
foreveryXandYinR
n
.Since
jx
i
Cy
i
jjx
i
jCjy
i
j;
summingbothsidesofthisequationfromi D1tonyields(8.1.4)withpD1.Tohandle
thecasewherep > 1,weneedthefollowinglemmas. . Theinequalityestablishedinthe
firstlemmaisknownasHölder’sinequality.
Lemma8.1.7
Supposethat
1
;
2
;...;
n
and
1
;
2
;...;
n
arenonnegativenumbers:
Letp>1andqDp=.p1/Ithus;
1
p
C
1
q
D1:
(8.1.5)
Then
Xn
iD1
i
i
Xn
iD1
p
i
!
1=p
Xn
iD1
q
i
!
1=q
:
(8.1.6)
Proof
Let˛andˇbeanytwopositivenumbers,andconsiderthefunction
f.ˇ/D
˛
p
p
C
ˇ
q
q
˛ˇ;
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