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492 Chapter7
IntegralsofFunctionsofSeveralVariables
Tocompletetheproof,wemustverify(7.3.15)foreveryrectangle
RDŒa
1
;b
1
Œa
2
;b
2
Œa
n
;b
n
DI
1
I
2
I
n
:
SupposethatAin(7.3.12)isanelementarymatrix;thatis,let
XDL.Y/DEY:
C
ASE
1.IfEisobtainedbyinterchangingtheithandjthrowsofI,then
x
r
D
8
<
:
y
r
ifr¤iandr¤jI
y
j
ifrDiI
y
i
ifrDj:
ThenL.R/istheCartesianproductofI
1
,I
2
,...,I
n
withI
i
andI
j
interchanged,so
V.L.R//DV.R/Djdet.E/jV.R/
sincedet.E/D1inthiscase(Exercise7.3.7
(a)
).
C
ASE
2.IfEisobtainedbymultiplyingtherthrowofIbya,then
x
r
D
y
r
ifr¤i;
ay
i
ifrDi:
Then
L.R/DI
1
I
i1
I
0
i
I
iC1
I
n
;
whereI
0
i
isanintervalwithlengthequaltojajtimesthelengthofI
i
,so
V.L.R//DjajV.R/Djdet.E/jV.R/
sincedet.E/Dainthiscase(Exercise7.3.7
(a)
).
C
ASE
x
r
D
y
r
ifr¤iI
y
i
Cay
j
ifrDi:
Then
L.R/D
˚
.x
1
;x
2
;:::;x
n
/
ˇ
ˇ
a
i
Cax
j
x
i
b
i
Cax
j
anda
r
x
r
b
r
ifr¤i
;
whichisaparallelogramifnD2andaparallelepipedifnD3(Figure7.3.1).Now
V.L.R//D
Z
L.R/
dX;
whichwecanevaluateasaniteratedintegralinwhichtheﬁrstintegrationiswithrespect
tox
i
.Forexample,ifi D1,then
V.L.R//D
Z
b
n
a
n
dx
n
Z
b
n1
a
n1
dx
n1

Z
b
2
a
2
dx
2
Z
b
1
Cax
j
a
1
Cax
j
dx
1
:
(7.3.19)
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Section7.3
ChangeofVariablesinMultipleIntegrals
493
Since
Z
b
1
Cax
j
a
1
Cax
j
dy
1
D
Z
b
1
a
1
dy
1
;
(7.3.19)canberewrittenas
V.L.R//D
Z
b
n
a
n
dx
n
Z
b
n1
a
n1
dx
n1

Z
b
2
a
2
dx
2
Z
b
1
a
1
dx
1
D.b
n
a
n
/.b
n1
a
n1
/.b
1
a
1
/DV.R/:
Hence,V.L.R//Djdet.E/jV.R/,sincedet.E/D1inthiscase(Exercise7.3.7
(a)
).
a
1
b
1
y
1
y
1
y
2
y
3
b
2
a
2
y
2
i = 1, j = 2, a > 0
i = 2, j = 3, a > 0
Figure7.3.1
Fromwhatwehaveshownsofar,(7.3.14)holdsifAisanelementarymatrixandSis
anycompactJordanmeasurableset.IfAisanarbitrarynonsingularmatrix,
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494 Chapter7
IntegralsofFunctionsofSeveralVariables
thenwecanwriteAasaproductofelementarymatrices(7.3.10)andapplyourknown
resultsuccessivelytoL
1
,L
2
,...,L
k
(see(7.3.13)).Thisyields
V.L.S//Djdet.E
k
/jjdet.E
k1
/jjdetE
1
jV.S/Djdet.A/jV.S/;
byTheorem6.1.9andinduction.
FormulationoftheRuleforChange ofVariables
Wenowformulatetheruleforchangeofvariablesinamultipleintegral.Sincewearefor
thepresentinterestedonlyin“discovering”therule,wewillmakeanyassumptionsthat
Throughouttherestofthissectionitwillbeconvenienttothinkoftherangeanddomain
ofatransformationGWR
n
!R
n
assubsetsofdistinctcopiesofR
n
.Wewilldenotethe
copycontainingD
G
asE
n
,andwriteGWE
n
!R
n
andXDG.Y/,reversingtheusual
rolesofXandY.
IfGisregularonasubsetSofE
n
,theneachXinG.S/canbeidentiﬁedbyspecifying
theuniquepointYinSsuchthatXDG.Y/.
Supposethatwewishtoevaluate
R
T
f.X/dX,whereTistheimageofacompactJordan
measurablesetSundertheregulartransformationXDG.Y/.Forsimplicity,wetakeSto
bearectangleandassumethatf iscontinuousonT DG.S/.
NowsupposethatP D D fR
1
;R
2
;:::;R
k
gisapartitionofS andT
j
D G.R
j
/(Fig-
ure7.3.2).
T
j
T
R
j
S
y
x
u
v
X = G(U)
Figure7.3.2
Then
Z
T
f.X/dXD
Xk
jD1
Z
T
j
f.X/dX
(7.3.20)
(Corollary7.1.31andinduction).Sincefiscontinuous,thereisapointX
j
inT
j
suchthat
Z
T
j
f.X/dXDf.X
j
/
Z
T
j
dXDf.X
j
/V.T
j
/
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Section7.3
ChangeofVariablesinMultipleIntegrals
495
(Theorem7.1.28),so(7.3.20)canberewrittenas
Z
T
f.X/dXD
Xk
jD1
f.X
j
/V.T
j
/:
(7.3.21)
NowweapproximateV.T
j
/.If
X
j
DG.Y
j
/;
(7.3.22)
thenY
j
2R
j
and,sinceGisdifferentiableatY
j
,
G.Y/G.Y
j
/CG
0
.Y
j
/.YY
j
/:
(7.3.23)
HereGandYY
j
arewrittenascolumnmatrices,G
0
means“approximatelyequal”inasensethatwecouldmakepreciseifwewished(Theo-
rem6.2.2).
ItisreasonabletoexpectthattheJordancontentofG.R
j
/isapproximatelyequaltothe
JordancontentofA.R
j
/,whereAistheafﬁnetransformation
A.Y/DG.Y
j
/CG
0
.Y
j
/.YY
j
/
ontherightsideof(7.3.23);thatis,
V.G.R
j
//V.A.R
j
//:
(7.3.24)
3
ıA
2
ıA
1
,where
A
1
.Y/DYY
j
;
A
2
.Y/DG
0
.Y
j
/Y;
and
A
3
.Y/DG.Y
j
/CY:
LetR
0
j
DA
1
.R
j
/.SinceA
1
merelyshiftsR
j
0
j
isalsoarectangle,
and
V.R
0
j
/DV.R
j
/:
(7.3.25)
NowletR
00
j
DA
2
.R
0
j
/.(Ingeneral,R
00
j
isnotarectangle.)SinceA
2
isthelineartransfor-
mationwithnonsingularmatrixG
0
.Y
j
/,Theorem7.3.7impliesthat
V.R
00
j
//DjdetG
0
.Y
j
/jV.R
0
j
/DjJG.Y
j
/jV.R
j
/;
(7.3.26)
whereJGistheJacobianofG.NowletR
000
j
DA
3
.R
00
j
/.SinceA
3
merelyshiftsallpoints
inthesameway,
V.R
000
j
/DV.R
00
j
/:
(7.3.27)
Now(7.3.24)–(7.3.27)suggestthat
V.T
j
/jJG.Y
j
/jV.R
j
/:
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496 Chapter7
IntegralsofFunctionsofSeveralVariables
(RecallthatT
j
DG.R
j
/.)Substitutingthisand(7.3.22)into(7.3.21)yields
Z
T
f.X/dX
Xk
jD1
f.G.Y
j
//jJG.Y
j
/jV.R
j
/:
ButthesumontherightisaRiemannsumfortheintegral
Z
S
f.G.Y//jJG.Y/jdY;
whichsuggeststhat
Z
T
f.X/dXD
Z
S
f.G.Y//jJG.Y/jdY:
Wewillprove thisbyan argumentthatwas publishedintheAmericanMathematical
Monthly[Vol.61(1954),pp.81-85]byJ.Schwartz.
TheMainTheorem
Wenowprovethefollowingformoftheruleforchangeofvariableinamultipleintegral.
Theorem7.3.8
SupposethatGWE!RisregularonacompactJordanmeasur-
ablesetSandf iscontinuousonG.S/:Then
Z
G.S/
f.X/dXD
Z
S
f.G.Y//jJG.Y/jdY:
(7.3.28)
Sincetheproofiscomplicated,webreakitdowntoaseriesoflemmas.Weﬁrstobserve
thatbothintegralsin(7.3.28)exist,byCorollary7.3.2,sincetheirintegrandsarecontinu-
ous.(NotethatSiscompactandJordanmeasurablebyassumption,andG.S/iscompact
andJordanmeasurablebyTheorem7.3.5.)Also,theresultistrivialifV.S/D0,sincethen
V.G.S//D0byLemma7.3.4,andbothintegralsin(7.3.28)vanish. Hence,weassume
thatV.S/>0.Weneedthefollowingdeﬁnition.
Deﬁnition7.3.9
ij
isannnmatrix;then
max
8
<
:
Xn
jD1
ja
ij
j
ˇ
ˇ
1in
9
=
;
istheinﬁnitynormofA;denotedbykAk
1
.
Lemma7.3.10
SupposethatGWE
n
!R
n
isregularonacubeCinE
n
;andletAbe
anonsingularnnmatrix:Then
V.G.C//jdet.A/j
max
˚
kA
1
G
0
.Y/k
1
ˇ
ˇ
Y2C

n
V.C/:
(7.3.29)
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Section7.3
ChangeofVariablesinMultipleIntegrals
497
Proof
LetsbetheedgelengthofC.LetY
0
D.c
1
;c
2
;:::;c
n
/bethecenterofC,and
supposethatHD.y
1
;y
2
;:::;y
n
/2C.IfHD.h
1
;h
2
;:::;h
n
/iscontinuouslydifferen-
tiableonC,thenapplyingthemeanvaluetheorem(Theorem5.4.5)tothecomponentsof
Hyields
h
i
.Y/h
i
.Y
0
/D
n
X
jD1
@h
i
.Y
i
/
@y
j
.y
j
c
j
/; 1in;
whereY
i
2C.Hence,recallingthat
H
0
.Y/D
@h
i
@y
j
n
i;jD1
;
applyingDeﬁnition7.3.9,andnotingthatjy
j
c
j
js=2,1j n,weinferthat
jh
i
.Y/h
i
.Y
0
/j
s
2
max
˚
kH
0
.Y/k
1
ˇ
ˇ
Y2C
; 1in:
ThismeansthatH.C/iscontainedinacubewithcenterX
0
DH.Y
0
/andedgelength
smax
˚
kH
0
.Y/k
1
ˇ
ˇ
Y2C
:
Therefore,
V.H.C//
maxfkH0.Y/k
1
n
ˇ
ˇ
Y2C
sn
D
maxfkH
0
.Y/k
1
n
ˇ
ˇ
Y2C
V.C/:
(7.3.30)
Nowlet
L.X/DA
1
X
andsetHDLıG;then
H.C/DL.G.C// and H
0
DA
1
G
0
;
so(7.3.30)impliesthat
V.L.G.C///
max
˚
kA
1
G
0
.Y/k
1
ˇ
ˇ
Y2C

n
V.C/:
(7.3.31)
SinceLislinear,Theorem7.3.7withAreplacedbyA
1
impliesthat
V.L.G.C///Djdet.A/
1
jV.G.C//:
Thisand(7.3.31)implythat
jdet.A
1
/jV.G.C//
max
˚
kA
1
G
0
.Y/k
1
ˇ
ˇ
Y2C

n
V.C/:
Sincedet.A
1
/D1=det.A/,thisimplies(7.3.29).
Lemma7.3.11
IfGWE
n
!R
n
isregularonacubeC inR
n
;then
V.G.C//
Z
C
jJG.Y/jdY:
(7.3.32)
498 Chapter7
IntegralsofFunctionsofSeveralVariables
Proof
LetPbeapartitionofCintosubcubesC
1
,C
2
,...,C
k
withcentersY
1
,Y
2
,...,
Y
k
.Then
V.G.C//D
Xk
jD1
V.G.C
j
//:
(7.3.33)
ApplyingLemma7.3.10toC
j
0
.A
j
/yields
V.G.C
j
//jJG.Y
j
/j
max
˚
k.G
0
.Y
j
//
1
G
0
.Y/k
1
ˇ
ˇ
Y2C
j

n
V.C
j
/:
(7.3.34)
Exercise6.1.22impliesthatif>0,thereisaı>0suchthat
max
˚
k.G
0
.Y
j
//
1
G
0
.Y/k
1
ˇ
ˇ
Y2C
j
<1C; 1j j k; if kPk<ı:
Therefore,from(7.3.34),
V.G.C
j
//.1C/
n
jJG.Y
j
/jV.C
j
/;
so(7.3.33)impliesthat
V.G.C//.1C/
n
Xk
jD1
jJG.Y
j
/jV.C
j
/ if kPk<ı:
SincethesumontherightisaRiemannsumfor
R
C
jJG.Y/jdYandcanbetakenarbi-
trarilysmall,thisimplies(7.3.32).
Lemma7.3.12
SupposethatS isJordanmeasurableand; > 0:Thenthereare
cubes C
1
;C
2
;...;C
r
inS withedgelengths< ;suchthatC
j
 S;1   j   r;
C
0
i
\C
0
j
D;ifi¤j;and
V.S/
Xr
jD1
V.C
j
/C:
(7.3.35)
Proof
SinceSisJordanmeasurable,
Z
C
S
.X/dXDV.S/
ifCisanycubecontainingS.Fromthisandthedeﬁnitionoftheintegral,thereisaı>0
suchthatifPisanypartitionofCwithkPk<ıandisanyRiemannsumof
S
over
P,then>V.S/=2.Therefore,ifs.P/isthelowersumof
S
overP,then
s.P/>V.S/ if kPk<ı:
(7.3.36)
NowsupposethatP D D fC
1
;C
2
;:::;C
k
g isapartitionofC intocubes s withkPk <
min.;ı/, andletC
1
, C
2
, ..., , C
k
benumberedsothatC
j
 S if1  j   r and
C
j
\Sc ¤ ¤ ;ifj > > r. From(7.3.4),s.P/ D
P
r
jD1
V.C
k
/. Thisand(7.3.36)imply
(7.3.35).Clearly,C
0
i
\C
0
j
D;ifi¤j.
Section7.3
ChangeofVariablesinMultipleIntegrals
499
Lemma7.3.13
SupposethatGWE
n
!R
n
isregularonacompactJordanmeasur-
ablesetSandf iscontinuousandnonnegativeonG.S/:Let
Q.S/D
Z
G.S/
f.X/dX
Z
S
f.G.Y//jJG.Y/jdY:
(7.3.37)
ThenQ.S/0:
Proof
FromthecontinuityofJGandf onthecompactsetsS S andG.S/, thereare
constantsM
1
andM
2
suchthat
jJG.Y/jM
1
if Y2S
(7.3.38)
and
jf.X/jM
2
if X2G.S/
(7.3.39)
(Theorem5.2.11). Nowsupposethat> > 0. . Sincef ıGisuniformlycontinuousonS
(Theorem5.2.14),thereisaı>0suchthat
jf.G.Y//f.G.Y
0
//j< if jYY
0
j<ıandY;Y
0
2S:
(7.3.40)
NowletC
1
,C
2
,...,C
r
bechosenasdescribedinLemma7.3.12,withDı=
p
n.Let
S
1
D
8
<
:
Y2S
ˇ
ˇ
Y…
r
[
jD1
C
j
9
=
;
:
ThenV.S
1
/<and
SD
0
@
[r
jD1
C
j
1
A
[S
1
:
(7.3.41)
SupposethatY
1
,Y
2
,...,Y
r
arepointsinC
1
,C
2
,...,C
r
andX
j
DG.Y
j
/,1j r.
From(7.3.41)andTheorem7.1.30,
Q.S/D
Z
G.S
1
/
f.X/dX
Z
S
1
f.G.Y//jJG.Y/jdY
C
Xr
jD1
Z
G.C
j
/
f.X/dX
Xr
jD1
Z
C
j
f.G.Y//jJG.Y/jdY
D
Z
G.S
1
/
f.X/dX
Z
S
1
f.G.Y//jJG.Y/jdY
C
Xr
jD1
Z
G.C
j
/
.f.X/f.A
j
//dX
C
r
X
jD1
Z
C
j
..f.G.Y
j
//f.G.Y///jJ.G.Y/jdY
C
r
X
jD1
f.X
j
/
V.G.C
j
//
Z
C
j
jJG.Y/jdY
!
: