pdfsharp c# : Add links to pdf document application software utility azure html asp.net visual studio TRENCH_REAL_ANALYSIS46-part266

452 Chapter7
IntegralsofFunctionsofSeveralVariables
Example7.1.6
Thefunction
f.x;y/D
(
xCy; 0x<y1;
5;
0yx1;
iscontinuousonRDŒ0;1Œ0;1exceptonthelinesegment
yDx; 0x1
(Figure7.1.5).Sincethelinesegmenthaszerocontent(Example7.1.5),fisintegrableon
R.
y
x
f(x, y) = x + y
f(x, y) = 5
y = x
1
1
Figure7.1.5
IntegralsoverMoreGeneralSubsets of
R
n
WecannowdefinetheintegralofaboundedfunctionovermoregeneralsubsetsofR
n
.
Definition7.1.17
Supposethatf isboundedonaboundedsubsetofSofR
n
,andlet
f
S
.X/D
(
f.X/; X2S;
0;
X62S:
(7.1.36)
LetRbearectanglecontainingS.Thentheintegraloff overSisdefinedtobe
Z
S
f.X/dXD
Z
R
f
S
.X/dX
if
R
R
f
S
.X/dXexists.
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Section7.1
DefinitionandExistenceoftheMultipleIntegral
453
Toseethatthisdefinitionmakessense,wemustshowthatifR
1
andR
2
aretworect-
anglescontainingSand
R
R
1
f
S
.X/dXexists,thensodoes
R
R
2
f
S
.X/dX,andthetwo
integralsareequal.TheproofofthisissketchedinExercise7.1.27.
Definition7.1.18
IfSisaboundedsubsetofR
n
andtheintegral
R
S
dX(withinte-
grandf 1)exists,wecall
R
S
dXthecontent(also,areaifnD2orvolumeifnD3)
ofS,anddenoteitbyV.S/;thus,
V.S/D
Z
S
dX:
Theorem7.1.19
Supposethatf isboundedonaboundedsetSandcontinuousex-
ceptonasubsetEofSwithzerocontent.Supposealsothat@Shaszerocontent:Thenf
isintegrableonS:
Proof
Letf
S
beasin(7.1.36).Sinceadiscontinuityoff
S
iseitheradiscontinuityoff
orapointof@S,thesetofdiscontinuitiesoff
S
istheunionoftwosetsofzerocontentand
thereforeisofzerocontent(Lemma7.1.15). Therefore,f
S
isintegrableonanyrectangle
containingS(fromTheorem7.1.16),andconsequentlyonS(Definition7.1.17).
DifferentiableSurfaces
Differentiablesurfaces,definedasfollows,formanimportantclassofsetsofzerocontent
inRn.
Definition7.1.20
AdifferentiablesurfaceSinR
n
.n>1/istheimageofacompact
subsetDofR
m
,wherem < n, , underacontinuouslydifferentiabletransformationG W
R
m
!R
n
.IfmD1,Sisalsocalledadifferentiablecurve.
Example7.1.7
Thecircle
˚
.x;y/
ˇ
ˇ
x
2
Cy
2
D9
isadifferentiablecurveinR
2
,sinceitistheimageofDDŒ0;2underthecontinuously
differentiabletransformationGWR!R
2
definedby
XDG./D
3cos
3sin
:
Example7.1.8
Thesphere
˚
.x;y;´/
ˇ
ˇ
x
2
Cy
2
2
D4
isadifferentiablesurfaceinR
3
,sinceitistheimageof
DD
˚
.;/
ˇ
ˇ
02;=2=2
underthecontinuouslydifferentiabletransformationGWR
2
!R
3
definedby
XDG.;/D
2
4
2coscos
2sincos
2sin
3
5
:
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454 Chapter7
IntegralsofFunctionsofSeveralVariables
Example7.1.9
Theset
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
x
i
0.iD1;2;3;4/;x
1
Cx
2
D1;x
3
Cx
4
D1
isadifferentiablesurfaceinR
4
, sinceitistheimageofD D D Œ0;1Œ0;1underthe
continuouslydifferentiabletransformationGWR
2
!R
4
definedby
XDG.u;v/D
2
6
6
4
u
1u
v
1v
3
7
7
5
:
Theorem7.1.21
AdifferentiablesurfaceinR
n
haszerocontent:
Proof
LetS, D, andG beas inDefinition7.1.20. From m Lemma6.2.7, thereisa
constantMsuchthat
jG.X/G.Y/jMjXYj if X;Y2D:
(7.1.37)
SinceDisbounded,Discontainedinacube
C DŒa
1
;b
1
Œa
2
;b
2
Œa
m
;b
m
;
where
b
i
a
i
DL; 1i i m:
SupposethatwepartitionC intoNsmallercubesbypartitioningeachoftheintervals
Œa
i
;b
i
intoNequalsubintervals.LetR
1
,R
2
,...,R
k
bethesmallercubessoproducedthat
containpointsofD,andselectpointsX
1
,X
2
,...,X
k
suchthatX
i
2D\R
i
,1ik.
IfY2D\R
i
,then(7.1.37)impliesthat
jG.X
i
/G.Y/jMjX
i
Yj:
(7.1.38)
SinceX
i
andYarebothinthecubeR
i
withedgelengthL=N,
jX
i
Yj
L
p
m
N
:
Thisand(7.1.38)implythat
jG.X
i
/G.Y/j
ML
p
m
N
;
whichinturnimpliesthatG.Y/liesinacube
e
R
i
inRcenteredatG.X
i
/,withsidesof
length2ML
p
m=N.Now
Xk
iD1
V.
e
R
i
/Dk
2ML
p
m
N
n
N
m
2ML
p
m
N
n
D.2ML
p
m/
n
N
mn
:
Sincen>m,wecanmakethesumontheleftarbitrarilysmallbytakingN sufficiently
large.Therefore,Shaszerocontent.
Theorems7.1.19and7.1.21implythefollowingtheorem.
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Section7.1
DefinitionandExistenceoftheMultipleIntegral
455
Theorem7.1.22
SupposethatSisaboundedsetinR
n
;withboundaryconsistingof
afinitenumberofdifferentiablesurfaces:Letf beboundedonSandcontinuousexcept
onasetofzerocontent.Thenf isintegrableonS:
Example7.1.10
Let
SD
˚
.x;y/
ˇ
ˇ
x
2
Cy
2
D1;x0
I
thus,S isboundedbyasemicircleandalinesegment(Figure7.1.6),bothdifferentiable
curvesinR2.Let
f.x;y/D
(
.1x
2
y
2
/
1=2
; .x;y/2S; ; y0;
.1x
2
y
2
/
1=2
; .x;y/2S; ; y<0:
ThenfiscontinousonSexceptonthelinesegment
yD0; 0x<1;
whichhaszerocontent,fromExample7.1.5. Hence, , Theorem 7.1.22impliesthatf is
integrableonS.
y
x
x2 + y2 = 1,  x ≥ 0
Figure7.1.6
PropertiesofMultipleIntegrals
Wenowlistsometheoremsonpropertiesofmultipleintegrals. Theproofsaresimilarto
thoseoftheanalogoustheoremsinSection3.3.
Note:BecauseofDefinition7.1.17,ifwesaythatafunctionf isintegrableonasetS,
thenSisnecessarilybounded.
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456 Chapter7
IntegralsofFunctionsofSeveralVariables
Theorem7.1.23
Iff andgareintegrableonS;thensoisf Cg;and
Z
S
.f Cg/.X/dXD
Z
S
f.X/dXC
Z
S
g.X/dX:
Proof
Exercise7.1.20.
Theorem7.1.24
Iff isintegrableonSandcisaconstant;thencf isintegrableon
S;and
Z
S
.cf/.X/dXDc
Z
S
f.X/dX:
Proof
Exercise7.1.21.
Theorem7.1.25
Iff andgareintegrableonSandf.X/g.X/forXinS;then
Z
S
f.X/dX
Z
S
g.X/dX:
Proof
Exercise7.1.22.
Theorem7.1.26
Iff isintegrableonS;thensoisjfj;and
ˇ
ˇ
ˇ
ˇ
Z
S
f.X/dX
ˇ
ˇ
ˇ
ˇ
Z
S
jf.X/jdX:
Proof
Exercise7.1.23.
Theorem7.1.27
Iff andgareintegrableonS;thensoistheproductfg:
Proof
Exercise7.1.24.
Theorem7.1.28
Supposethatuiscontinuousandvisintegrableandnonnegativeon
arectangleR:Then
Z
R
u.X/v.X/dXDu.X
0
/
Z
R
v.X/dX
forsomeX
0
inR:
Proof
Exercise7.1.25.
Lemma7.1.29
SupposethatS iscontainedinaboundedsetT andf isintegrable
onS:Thenf
S
.see(7.1.36)/isintegrableonT;and
Z
T
f
S
.X/dXD
Z
S
f.X/dX:
Proof
FromDefinition7.1.17withf andSreplacedbyf
S
andT,
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Section7.1
DefinitionandExistenceoftheMultipleIntegral
457
.f
S
/
T
.X/D
f
S
.X/; X2T;
0;
X62T:
SinceS  T,.f
S
/
T
D f
S
. (Verify.) NowsupposethatRisarectanglecontainingT.
ThenRalsocontainsS(Figure7.1.7),
R
T
Figure7.1.7
so
Z
S
f.X/dXD
Z
R
f
S
.X/dX
(Definition7.1.17,appliedtofandS/
D
Z
R
.f
S
/
T
.X/dX (since.f
S
/
T
Df
S
)
D
Z
T
f
S
.X/dX
(Definition7.1.17,appliedtof
S
andT/;
whichcompletestheproof.
Theorem7.1.30
Iff isintegrableondisjointsetsS
1
andS
2
;thenf isintegrableon
S
1
[S
2
;and
Z
S
1
[S
2
f.X/dXD
Z
S
1
f.X/dXC
Z
S
2
f.X/dX:
(7.1.39)
Proof
ForiD1,2,let
f
S
i
.X/D
(
f.X/; X2S
i
;
0;
X62S
i
:
FromLemma7.1.29withSDS
i
andT DS
1
[S
2
,f
S
i
isintegrableonS
1
[S
2
,and
Z
S
1
[S
2
f
S
i
.X/dXD
Z
S
i
f.X/dX; iD1;2:
Theorem7.1.23nowimpliesthatf
S
1
Cf
S
2
isintegrableonS
1
[S
2
and
Z
S
1
[S
2
.f
S
1
Cf
S
2
/.X/dXD
Z
S
1
f.X/dXC
Z
S
2
f.X/dX:
(7.1.40)
458 Chapter7
IntegralsofFunctionsofSeveralVariables
SinceS
1
\S
2
D;,
f
S
1
Cf
S
2
.X/Df
S
1
.X/Cf
S
2
.X/Df.X/; X2S
1
[S
2
:
Therefore,(7.1.40)implies(7.1.39).
WeleaveittoyoutoprovethefollowingextensionofTheorem7.1.30.(Exercise7.1.31
(b)
).
Corollary7.1.31
Supposethatf isintegrableonsetsS
1
andS
2
suchthatS
1
\S
2
haszerocontent:Thenf isintegrableonS
1
[S
2
;and
Z
S
1
[S
2
f.X/dXD
Z
S
1
f.X/dXC
Z
S
2
f.X/dX:
Example7.1.11
Let
S
1
D
˚
.x;y/
ˇ
ˇ
0x1;0y1Cx
and
S
2
D
˚
.x;y/
ˇ
ˇ
1x0;0y1x
(Figure7.1.8).
S
y
x
y = 1 − x
y = 1 + x
1
−1
Figure7.1.8
Then
S
1
\S
2
D
˚
.0;y/
ˇ
ˇ
0y1
haszerocontent.Hence,Corollary7.1.31impliesthatiffisintegrableonS
1
andS
2
,then
f isalsointegrableover
SDS
1
[S
2
D
˚
.x;y/
ˇ
ˇ
1x1;0y1Cjxj
(Figure7.1.9),and
Z
S
1
[S
2
f.X/dXD
Z
S
1
f.X/dXC
Z
S
2
f.X/dX:
Section7.1
DefinitionandExistenceoftheMultipleIntegral
459
y
y
x
x
y = 1 − x
y = 1 + x
S
1
S
2
Figure7.1.9
Wewilldiscussthisexamplefurtherinthenextsection.
7.1Exercises
1.
Prove:IfRisdegenerate,thenDefinition7.1.2impliesthat
R
R
f.X/dXD0iff
isboundedonR.
2.
EvaluatedirectlyfromDefinition7.1.2.
(a)
R
R
.3xC2y/d.x;y/; RDŒ0;2Œ1;3
(b)
R
R
xyd.x;y/; RDŒ0;1Œ0;1
3.
Supposethat
R
b
a
f.x/dxand
R
d
c
g.y/dyexist,andletRDŒa;bŒc;d.Criticize
thefollowing“proof”that
R
R
f.x/g.y/d.x;y/existsandequals
Z
b
a
f.x/dx
Z
d
c
g.y/dy
!
:
(SeeExercise7.1.30foracorrectproofofthisassertion.)
“Proof.”Let
P
1
WaDx
0
<x
1
<<x
r
Db and P
2
WcDy
0
<y
1
<<y
s
Dd
bepartitionsofŒa;bandŒc;d,andP DP
1
P
2
.ThenatypicalRiemannsumof
fgoverPisoftheform
D
Xr
iD1
Xs
jD1
f.
i
/g.
j
/.x
i
x
i1
/.y
j
y
j1
/D
1
2
;
where
1
D
Xr
iD1
f.
i
/.x
i
x
i1
/ and 
2
D
Xs
jD1
g.
j
/.y
j
y
j1
/
460 Chapter7
IntegralsofFunctionsofSeveralVariables
aretypicalRiemannsumsoff overŒa;bandgoverŒc;d. . Sincef andgare
integrableontheseintervals,
ˇ
ˇ
ˇ
ˇ
ˇ
1
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
and
ˇ
ˇ
ˇ
ˇ
ˇ
2
Z
d
c
g.y/dy
ˇ
ˇ
ˇ
ˇ
ˇ
canbemadearbitrarilysmallbytakingkP
1
kandkP
2
ksufficientlysmall. From
this,itisstraightforwardtoshowthat
ˇ
ˇ
ˇ
ˇ
ˇ

Z
b
a
f.x/dx
Z
d
c
g.y/dy
!
ˇ
ˇ
ˇ
ˇ
ˇ
canbemadearbitrarilysmallbytakingkPksufficientlysmall. Thisimpliesthe
statedresult.
4.
Supposethatf.x;y/  0onR D Œa;bŒc;d. Justifytheinterpretationof
R
R
f.x;y/d.x;y/, ifitexists,asthevolumeoftheregioninR
3
boundedbythe
surfaces´Df.x;y/andtheplanes´D0,xDa,xDb,yDc,andyDd.
5.
ProveTheorem7.1.5.H
INT
:SeetheproofofTheorem3.1.4:
6.
Supposethat
f.x;y/D
8
ˆ
ˆ
<
ˆ
ˆ
:
0
ifxandyarerational,
1
ifxisrationalandyisirrational,
2
ifxisirrationalandyisrational,
3
ifxandyareirrational.
Find
Z
R
f.x;y/d.x;y/ and
Z
R
f.x;y/d.x;y/ if RDŒa;bŒc;d:
7.
ProveEqn.(7.1.17)ofLemma7.1.6.
8.
ProveTheorem7.1.7H
INT
:SeetheproofofTheorem3.2.2:
9.
ProveTheorem7.1.8H
INT
:SeetheproofofTheorem3.2.3:
10.
ProveLemma7.1.9H
INT
:SeetheproofofLemma3.2.4:
11.
ProveTheorem7.1.10H
INT
:SeetheproofofTheorem3.2.5:
12.
ProveTheorem7.1.12H
INT
:SeetheproofofTheorem3.2.7:
13.
GiveanexampleofadenumerablesetinR2thatdoesnothavezerocontent.
14.
Prove:
(a)
IfS
1
andS
2
havezerocontent,thenS
1
[S
2
haszerocontent.
(b)
IfS
1
haszerocontentandS
2
S
1
,thenS
2
haszerocontent.
(c)
IfShaszerocontent,then
Shaszerocontent.
15.
Showthatadegeneraterectanglehaszerocontent.
Section7.1
DefinitionandExistenceoftheMultipleIntegral
461
16.
Supposethatf iscontinuousonacompactsetS S inR
n
. Showthatthesurface
´Df.X/,X2S,haszerocontentinR
nC1
.H
INT
:SeeExample7.1.5:
17.
LetSbeaboundedsetsuchthatS\@Sdoesnothavezerocontent.
(a)
Supposethatf isdefinedonSandf.X/>0onasubsetTofS\@S
thatdoesnothavezerocontent.Showthatf isnotintegrableonS.
(b)
ConcludethatV.S/isundefined.
18. (a)
Supposethathisboundedandh.X/ D D 0exceptonasetofzerocontent.
Showthat
R
S
h.X/dXD0foranyboundedsetS.
(b)
Supposethat
R
S
f.X/dXexists,gisboundedonS,andf.X/Dg.X/except
forXinasetofzerocontent.ShowthatgisintegrableonSand
Z
S
g.X/dXD
Z
S
f.X/dX:
19.
Supposethatf isintegrableonasetSandS
0
isasubsetofSsuchthat@S
0
has
zerocontent.Showthatf isintegrableonS
0
.
20.
ProveTheorem7.1.23H
INT
:SeetheproofofTheorem3.3.1:
21.
ProveTheorem7.1.24.
22.
ProveTheorem7.1.25H
INT
:SeetheproofofTheorem3.3.4:
23.
ProveTheorem7.1.26H
INT
:SeetheproofofTheorem3.3.5:
24.
ProveTheorem7.1.27H
INT
:SeetheproofofTheorem3.3.6:
25.
ProveTheorem7.1.28H
INT
:SeetheproofofTheorem3.3.7:
26.
Prove:Iff isintegrableonarectangleR,thenfisintegrableonanysubrectangle
ofR.H
INT
:UseTheorem7.1.12IseetheproofofTheorem3.3.8:
27.
SupposethatRand
e
Rarerectangles,R
e
R,gisboundedon
e
R,andg.X/D0if
X62R.
(a)
Showthat
R
e
R
g.X/dXexistsifandonlyif
R
R
g.X/dXexistsand,inthis
case,
Z
e
R
g.X/dXD
Z
R
g.X/dX:
H
INT
:UseExercise7.1.26:
(b)
Use
(a)
toshowthatDefinition7.1.17islegitimate;thatis,theexistenceand
valueof
R
S
f.X/dXdoesnotdependontheparticularrectanglechosento
containS.
28. (a)
Supposethatf isintegrableonarectangleRandP DfR
1
;R
2
;:::;R
k
gis
apartitionofR.Showthat
Z
R
f.X/dXD
Xk
jD1
Z
R
j
f.X/dX:
H
INT
:UseExercise7.1.26:
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