432 Chapter6
Vector-ValuedFunctionsofSeveralVariables
7.
Findu.x
0
;y
0
/,u
x
.x
0
;y
0
/,andu
y
.x
0
;y
0
/forallcontinuouslydifferentiablefunc-
tionsuthatsatisfythegivenequationnear.x
0
;y
0
/.
(a)
2x2y43uxyCu2x4yD0; .x
0
;y
0
/D.1;1/
(b)
cosucosxCsinusinyD0; .x
0
;y
0
/D.0;/
8.
SupposethatUD.u;v/iscontinuouslydifferentiablewithrespectto.x;y;´/and
satisfies
x
2
C4y
2
2
2u
2
Cv
2
D4
.xC´/
2
CuvD3
and
u.1;
1
2
;1/D2; v.1;
1
2
;1/D1:
FindU
0
.1;
1
2
;1/.
9.
Letuandvbecontinuouslydifferentiablewithrespecttoxandsatisfy
uC2u
2
Cv
2
Cx
2
C2vxD0
xuvCe
u
sin.vCx/D0
andu.0/Dv.0/D0.Findu0.0/andv0.0/.
10.
LetUD.u;v;w/becontinuouslydifferentiablewithrespectto.x;y/andsatisfy
x2yCxyCu2.vCw/2D3
exCy uvwD2
.xCy/
2
CuCvCw
2
D
3
andU.1;1/D.1;2;0/.FindU
0
.1;1/.
11.
TwocontinuouslydifferentiabletransformationsU D D .u;v/of.x;y/satisfythe
system
xyu4yuC9xvD0
2xy3y
2
Cv
2
D0
near.x
0
;y
0
/ D .1;1/. Findthevalueofeachtransformationanditsdifferential
matrixat.1;1/.
12.
Supposethatu,v,andwarecontinuouslydifferentiablefunctionsof.x;y;´/that
satisfythesystem
e
x
cosyCe
´
cosuCe
v
coswCxD3
e
x
sinyCe
´
sinuCe
v
cosw
D1
e
x
tanyCe
´
tanuCe
v
tanwC´D0
near.x
0
;y
0
0
/ D.0;0;0/,andu.0;0;0/D v.0;0;0/Dw.0;0;0/D 0. Find
u
x
.0;0;0/,v
x
.0;0;0/,andw
x
.0;0;0/.
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Section6.4
TheImplicitFunctionTheorem
433
13.
LetFD.f;g;h/becontinuouslydifferentiableinaneighborhoodofP
0
D.x
0
;y
0
0
;u
0
;v
0
/,
F.P
0
/D0,and
@.f;g;h/
@.y;´;u/
ˇ
ˇ
ˇ
ˇ
P
0
¤0:
ThenTheorem6.4.1impliesthattheconditions
F.x;y;´;u;v/D0; y.x
0
;v
0
/Du
0
; ´.x
0
;v
0
/D´
0
; u.x
0
;v
0
/Du
0
determiney,´,anduascontinuouslydifferentiablefunctionsof.x;v/near.x
0
;v
0
/.
UseCramer’sruletoexpresstheirfirstpartialderivativesasratiosofJacobians.
14.
Decidewhichpairsofthevariablesx,y,´,u,andvaredeterminedasfunctionsof
theothersbythesystem
xC2yC3´C uC6vD0
2xC4yC ´C2uC2vD0;
andsolveforthem.
15.
Letyandvbecontinuouslydifferentiablefunctionsof.x;´;u/thatsatisfy
x
2
C4y
2
2
2u
2
Cv
2
D4
.xC´/
2
CuvD3
near.x
0
0
;u
0
/D.1;1;2/,andsupposethat
y.1;1;2/D
1
2
; v.1;1;2/D1:
Findy
x
.1;1;2/andv
u
.1;1;2/.
16.
Letu,v,andxbecontinuouslydifferentiablefunctionsof.w;y/thatsatisfy
x
2
yCxy
2
Cu
2
.vCw/
2
D3
e
xCy
uvwD2
.xCy/
2
CuCvCw
2
D
3
near.w
0
;y
0
/D.0;1/,andsupposethat
u.0;1/D1; v.0;1/D2; x.0;1/D1:
Findthefirstpartialderivativesofu,v,andxwithrespecttoyandwat.0;1/.
17.
Inadditiontotheassumptions ofTheorem 6.4.1, supposethatFhasallpartial
derivativesoforder q q inS. ShowthatU U D U.X/hasallpartialderivatives
oforderqinN.
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434 Chapter6
Vector-ValuedFunctionsofSeveralVariables
18.
Calculateallfirstandsecondpartialderivativesat.x
0
;y
0
/D.1;1/ofthefunctions
uandvthatsatisfy
x
2
Cy
2
Cu
2
Cv
2
D3
x Cy Cu Cv D3;
u.1;1/D0; v.1;1/D1:
19.
Calculateallfirstandsecondpartialderivativesat.x
0
;y
0
/D.1;1/ofthefunc-
tionsuandvthatsatisfy
u
2
v
2
Dxy2
2uvDxCy2;
u.1;1/D1; v.1;1/D1:
20.
Supposethatf
1
, f
2
, ..., f
n
arecontinuouslydifferentiablefunctionsofXina
regionSinR
n
,iscontinuouslydifferentiablefunctionofUinaregionT ofR
n
,
.f
1
.X/;f
2
.X/;:::;f
n
.X//2T; X2S;
.f
1
.X/;f
2
.X/;:::;f
n
.X//D0; X2S;
and
Xn
jD1
2
u
j
.U/>0; U2T:
Showthat
@.f
1
;f
2
;:::;f
n
/
@.x
1
;x
2
;:::;x
n
/
D0; X2S:
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CHAPTER7
IntegralsofFunctions
ofSeveralVariables
INTHISCHAPTERwestudytheintegralcalculus ofreal-valuedfunctionsofseveral
variables.
SECTION7.1definesmultipleintegrals,firstoverrectangularparallelepipedsinR
n
and
thenovermoregeneralsets.Thediscussiondealswiththemultipleintegralofafunction
whosediscontinuitiesformasetofJordancontentzero,overasetwhoseboundaryhas
Jordancontentzero.
SECTION7.2dealswithevaluationofmultipleintegralsbymeansofiteratedintegrals.
SECTION7.3beginswiththedefinitionofJordanmeasurability,followedbyaderivation
oftheruleforchangeofcontentunderalineartransformation,anintuitiveformulationof
theruleforchangeofvariablesinmultipleintegrals,andfinallyacarefulstatementand
proofoftherule.Thisisacomplicatedproof.
7.1 DEFINITION AND EXISTENCEOF F THEMULTIPLEIN-
TEGRAL
WenowconsidertheRiemannintegralofareal-valuedfunctionf definedonasubsetof
R
n
, wheren n   2. Muchofthisdevelopmentwillbeanalogoustothedevelopmentin
Sections3.1–3fornD1,butthereisanimportantdifference:fornD1,weconsidered
integralsoverclosedintervalsonly, butforn n > 1wemustconsidermorecomplicated
regionsofintegration.Todefercomplicationsduetogeometry,wefirstconsiderintegrals
overrectanglesinR
n
,whichwenowdefine.
IntegralsoverRectangles
The
S
1
S
2
S
n
ofsubsetsS
1
,S
2
,...,S
n
ofRisthesetofpoints.x
1
;x
2
;:::;x
n
/inR
n
suchthatx
1
2
S
1
;x
2
2S
2
;:::;x
n
2S
n
.Forexample,theCartesianproductofthetwoclosedintervals
435
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436 Chapter7
IntegralsofFunctionsofSeveralVariables
Œa
1
;b
1
Œa
2
;b
2
D
˚
.x;y/
ˇ
ˇ
a
1
xb
1
;a
2
yb
2
isarectangleinR
2
withsidesparalleltothex-andy-axes(Figure7.1.1).
y
x
a
1
b
1
a
2
b
2
Figure7.1.1
TheCartesianproductofthreeclosedintervals
Œa
1
;b
1
Œa
2
;b
2
Œa
3
;b
3
D
˚
.x;y;´/
ˇ
ˇ
a
1
xb
1
;a
2
yb
2
; a
3
´b
3
isarectangularparallelepipedinR3withfacesparalleltothecoordinateaxes(Figure7.1.2).
z
y
x
Figure7.1.2
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Section7.1
DefinitionandExistenceoftheMultipleIntegral
437
Definition7.1.1
AcoordinaterectangleRinR
n
istheCartesianproductofnclosed
intervals;thatis,
RDŒa
1
;b
1
Œa
2
;b
2
Œa
n
;b
n
:
ThecontentofRis
V.R/D.b
1
a
1
/.b
2
a
2
/.b
n
a
n
/:
Thenumbersb
1
a
1
,b
2
a
2
,...,b
n
a
n
aretheedgelengthsofR. Iftheyareequal,
thenRisacoordinatecube. Ifa
r
Db
r
forsomer,thenV.R/D0andwesaythatRis
degenerate;otherwise,Risnondegenerate.
Ifn D D 1,2,or3,thenV.R/is,respectively, thelengthofaninterval,theareaofa
rectangle,orthevolumeofarectangularparallelepiped.Henceforth,“rectangle”or“cube”
willalwaysmean“coordinaterectangle”or“coordinatecube”unlessitisstatedotherwise.
If
RDŒa
1
;b
1
Œa
2
;b
2
Œa
n
;b
n
and
P
r
Wa
r
Da
r0
<a
r1
<<a
rm
r
Db
r
isapartitionofŒa
r
;b
r
,1rn,thenthesetofallrectanglesinR
n
thatcanbewritten
as
Œa
1;j
1
1
;a
1j
1
Œa
2;j
2
1
;a
2j
2
Œa
n;j
n
1
;a
nj
n
; 1j
r
m
r
; 1rn;
isapartitionofR.Wedenotethispartitionby
P DP
1
P
2
P
n
(7.1.1)
anddefineitsnormtobethemaximumofthenormsofP
1
, P
2
, ..., P
n
,asdefinedin
Section3.1;thus,
kPkDmaxfkP
1
k;kP
2
k;:::;kP
n
kg:
Putanotherway,kPkisthelargestoftheedgelengthsofallthesubrectanglesinP.
Geometrically,arectangleinR
2
ispartitionedbydrawinghorizontalandverticallines
throughit(Figure7.1.3);inR
3
,bydrawingplanesthroughitparalleltothecoordinateaxes.
PartitioningdividesarectangleRintofinitelymanysubrectanglesthatwecannumberin
arbitraryorderasR
1
,R
2
,...,R
k
.Sometimesitisconvenienttowrite
P DfR
1
;R
2
;:::;R
k
g
ratherthan(7.1.1).
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438 Chapter7
IntegralsofFunctionsofSeveralVariables
y
x
a
1
b
1
a
2
b
2
Figure7.1.3
IfP DP
1
P
2
P
n
andP
0
DP
0
1
P
0
2
P
0
n
arepartitionsofthesame
rectangle,thenP
0
isarefinementofP ifP
0
i
isarefinementofP
i
,1i n,asdefinedin
Section3.1.
Supposethatfisareal-valuedfunctiondefinedonarectangleRinR
n
,PDfR
1
;R
2
;:::;R
k
g
isapartitionofR,andX
j
isanarbitrarypointinR
j
,1j k.Then
D
Xk
jD1
f.X
j
/V.R
j
/
isaRiemannsumoffoverP.SinceX
j
canbechosenarbitrarilyinR
j
,thereareinfinitely
manyRiemannsumsforagivenfunctionf overanypartitionP P ofR.
ThefollowingdefinitionissimilartoDefinition3.1.1.
Definition7.1.2
Letf beareal-valuedfunctiondefinedonarectangleRinR
n
. We
saythatf isRiemannintegrableonRifthereisanumberLwiththefollowingproperty:
Forevery>0,thereisaı>0suchthat
jLj<
ifisanyRiemannsumoff overapartitionPofRsuchthatkPk<ı.Inthiscase,we
saythatListheRiemannintegraloff overR,andwrite
Z
R
f.X/dXDL:
IfRisdegenerate,thenDefinition7.1.2impliesthat
R
R
f.X/dXD0foranyfunctionf
definedonR(Exercise7.1.1).Therefore,itshouldbeunderstoodhenceforththatwhenever
wespeakofarectangleinR
n
wemeananondegeneraterectangle,unlessitisstatedtothe
contrary.
Section7.1
DefinitionandExistenceoftheMultipleIntegral
439
Theintegral
R
R
f.X/dXisalsowrittenas
Z
R
f.x;y/d.x;y/ .nD2/;
Z
R
f.x;y;´/d.x;y;´/ .nD3/;
or
Z
R
f.x
1
;x
2
;:::;x
n
/d.x
1
;x
2
;:::;x
n
/ (narbitrary):
HeredXdoesnotstandforthedifferentialofX,asdefinedinSection6.2. Itmerely
identifiesx
1
,x
2
,...,x
n
,thecomponentsofX,asthevariablesofintegration.Toavoidthis
minorinconsistency,someauthorswritesimply
R
R
f ratherthan
R
R
f.X/dX.
Asinthecasewheren D D 1,wewillsaysimply“integrable”or“integral”whenwe
mean“Riemannintegrable”or“Riemannintegral.”Ifn2,wecalltheintegralofDefi-
nition7.1.2amultipleintegral;fornD2andnD3wealsocallthemdoubleandtriple
integrals,respectively. Whenwewishtodistinguishbetweenmultipleintegralsandthe
integralwestudiedinChapter .nD1/,wewillcallthelatteranordinaryintegral.
Example7.1.1
Find
R
R
f.x;y/d.x;y/,where
RDŒa;bŒc;d
and
f.x;y/DxCy:
Solution
LetP
1
andP
2
bepartitionsofŒa;bandŒc;d;thus,
P
1
WaDx
0
<x
1
<<x
r
Db and
P
2
WcDy
0
<y
1
<<y
s
Dd:
AtypicalRiemannsumoffoverP DP
1
P
2
isgivenby
D
Xr
iD1
Xs
jD1
.
ij
C
ij
/.x
i
x
i1
/.y
j
y
j1
/;
(7.1.2)
where
x
i1

ij
x
i
and y
j1

ij
y
j
:
(7.1.3)
ThemidpointsofŒx
i1
;x
i
andŒy
j1
;y
j
are
x
i
D
x
i
Cx
i1
2
and
y
j
D
y
j
Cy
j1
2
;
(7.1.4)
and(7.1.3)impliesthat
j
ij
x
i
j
x
i
x
i1
2
kP
1
k
2
kPk
2
(7.1.5)
and
j
ij
y
j
j
y
j
y
j1
2
kP
2
k
2
kPk
2
:
(7.1.6)
440 Chapter7
IntegralsofFunctionsofSeveralVariables
Nowwerewrite(7.1.2)as
D
Xr
iD1
Xs
jD1
.
x
i
C
y
j
/.x
i
x
i1
/.y
j
y
j1
/
C
Xr
iD1
Xs
jD1
.
ij
x
i
/C.
ij
y
j
/
.x
i
x
i1
/.y
j
y
j1
/:
(7.1.7)
Tofind
R
R
f.x;y/d.x;y/from(7.1.7),werecallthat
Xr
iD1
.x
i
x
i1
/Dba;
Xs
jD1
.y
j
y
j1
/Ddc
(7.1.8)
(Example3.1.1),and
Xr
iD1
.x
2
i
x
2
i1
/Db
2
a
2
;
Xs
jD1
.y
2
j
y
2
j1
/Dd
2
c
2
(7.1.9)
(Example3.1.2).
Becauseof(7.1.5)and(7.1.6)theabsolutevalueofthesecondsumin(7.1.7)doesnot
exceed
kPk
r
X
jD1
s
X
jD1
.x
i
x
i1
/.y
j
y
j1
/DkPk
"
r
X
iD1
.x
i
x
i1
/
#
2
4
s
X
jD1
.y
j
y
j1
/
3
5
DkPk.ba/.dc/
(see(7.1.8)),so(7.1.7)impliesthat
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ

Xr
iD1
Xs
jD1
.
x
i
C
y
j
/.x
i
x
i1
/.y
j
y
j1
/
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
kPk.ba/.dc/:
(7.1.10)
Itnowfollowsthat
Xr
iD1
Xs
jD1
x
i
.x
i
x
i1
/.y
j
y
j1
/D
"
Xr
iD1
x
i
.x
i
x
i1
/
#
2
4
Xs
jD1
.y
j
y
j1
/
3
5
D.dc/
r
X
iD1
x
i
.x
i
x
i1
/ (from(7.1.8))
D
dc
2
Xr
iD1
.x
2
i
x
2
i1
/
(from(7.1.4))
D
dc
2
.b
2
a
2
/
(from(7.1.9)):
Similarly,
Xr
iD1
Xs
jD1
y
j
.x
i
x
i1
/.y
j
y
j1
/D
ba
2
.d
2
c
2
/:
Section7.1
DefinitionandExistenceoftheMultipleIntegral
441
Therefore,(7.1.10)canbewrittenas
ˇ
ˇ
ˇ
ˇ

dc
2
.b
2
a
2
/
ba
2
.d
2
c
2
/
ˇ
ˇ
ˇ
ˇ
kPk.ba/.dc/:
SincetherightsidecanbemadeassmallaswewishbychoosingkPksufficientlysmall,
Z
R
.xCy/d.x;y/D
1
2
.dc/.b
2
a
2
/C.ba/.d
2
c
2
/
:
UpperandLowerIntegrals
ThefollowingtheoremisanalogoustoTheorem3.1.2.
Theorem7.1.3
Iff isunboundedonthenondegeneraterectangleRinR
n
;thenf is
notintegrableonR:
Proof
Wewillshowthatiff isunboundedonR,P P DfR
1
;R
2
;:::;R
k
gisanyparti-
tionofR,andM>0,thenthereareRiemannsumsand
0
off overPsuchthat
j
0
jM:
(7.1.11)
Thisimpliesthatf cannotsatisfyDefinition7.1.2.(Why?)
Let
D
Xk
jD1
f.X
j
/V.R
j
/
beaRiemannsumoff overP.Theremustbeanintegeriinf1;2;:::;kgsuchthat
jf.X/f.X
i
/j
M
V.R
i
/
(7.1.12)
forsomeXinR
i
,becauseifthiswerenotso,wewouldhave
jf.X/f.X
j
/j<
M
V.R
j
/
; X2R
j
;
1j k:
Ifthisisso,then
jf.X/jDjf.X
j
/Cf.X/f.X
j
/jjf.X
j
/jCjf.X/f.X
j
/j
jf.X
j
/jC
M
V.R
j
/
; X2R
j
; 1j j k:
However,thisimpliesthat
jf.X/jmax
jf.X
j
/jC
M
V.R
j
/
ˇ
ˇ
1jk
; X2R;
whichcontradictstheassumptionthatf isunboundedonR.
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