c# pdf free : Add links to pdf document control application system web page html windows console TRENCH_REAL_ANALYSIS34-part253

332 Chapter5
Real-ValuedFunctionsofSeveralVariables
lim
.x;y/!.x
0
;y
0
/
f.x;y/T.x;y/
p
.xx
0
/2C.yy
0
/2
D0
(Figure5.3.2). Moreover, , (5.3.28)istheonlyplaneinR
3
withtheseproperties(Exer-
cise5.3.25). Wesaythatthisplaneistangenttothesurface´ ´ D f.x;y/atthepoint
.x
0
;y
0
;f.x
0
;y
0
//. Wewillnowshowthatitisthe“limit”of“secantplanes”associated
withthesurface´Df.x;y/,justasatangentlinetoacurveyDf.x/inR
3
isthelimit
ofsecantlinestothecurve(Section2.3).
LetX
i
D.x
i
;y
i
/.i D1;2;3/. . Theequationofthe“secantplane”throughthepoints
.x
i
;y
i
;f.x
i
;y
i
//.i D1;2;3/onthesurface´Df.x;y/(Figure5.3.3)isoftheform
´Df.X
0
/CA.xx
0
/CB.yy
0
/;
(5.3.29)
whereAandBsatisfythesystem
f.X
1
/Df.X
0
/CA.x
1
x
0
/CB.y
1
y
0
/;
f.X
2
/Df.X
0
/CA.x
2
x
0
/CB.y
2
y
0
/:
SolvingforAandByields
AD
.f.X
1
/f.X
0
//.y
2
y
0
/.f.X
2
/f.X
0
//.y
1
y
0
/
.x
1
x
0
/.y
2
y
0
/.x
2
x
0
/.y
1
y
0
/
(5.3.30)
and
BD
.f.X
2
/f.X
0
//.x
1
x
0
/.f.X
1
/f.X
0
//.x
2
x
0
/
.x
1
x
0
/.y
2
y
0
/.x
2
x
0
/.y
1
y
0
/
(5.3.31)
if
.x
1
x
0
/.y
2
y
0
/.x
2
x
0
/.y
1
y
0
/¤0;
(5.3.32)
whichisequivalenttotherequirementthatX
0
,X
1
, andX
2
donotlieonaline(Exer-
cise5.3.23).Ifwewrite
X
1
DX
0
CtU and X
2
DX
0
CtV;
whereU D .u
1
;u
2
/andV D D .v
1
;v
2
/are fixednonzerovectors (Figure5.3.3), then
(5.3.30),(5.3.31),and(5.3.32)takethemoreconvenientforms
AD
f.X
0
CtU/f.X
0
/
t
v
2
f.X
0
CtV/f.X
0
/
t
u
2
u
1
v
2
u
2
v
1
;
(5.3.33)
BD
f.X
0
CtV/f.X
0
/
t
u
1
f.X
0
CtU/f.X
0
/
t
v
1
u
1
v
2
u
2
v
1
;
(5.3.34)
and
u
1
v
2
u
2
v
1
¤0:
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Section5.3
PartialDerivativesandtheDifferential
333
y
z
x
X
0
X
2
X
1
V
U
Figure5.3.3
IffisdifferentiableatX
0
,then
f.X/f.X
0
/Df
x
.X
0
/.xx
0
/Cf
y
.X
0
/.yy
0
/C.X/jXX
0
j;
(5.3.35)
where
lim
X!X
0
.X/D0:
(5.3.36)
SubstitutingfirstXDX
0
CtUandthenXDX
0
CtVin(5.3.35)anddividingbytyields
f.X
0
CtU/f.X
0
/
t
Df
x
.X
0
/u
1
Cf
y
.X
0
/u
2
CE
1
.t/jUj
(5.3.37)
and
f.X
0
CtV/f.X
0
/
t
Df
x
.X
0
/v
1
Cf
y
.X
0
/v
2
CE
2
.t/jVj;
(5.3.38)
where
E
1
.t/D.X
0
CtU/jtj=t and E
2
.t/D.X
0
CtV/jtj=t;
so
lim
t!0
E
i
.t/D0; iD1;2;
(5.3.39)
becauseof(5.3.36).Substituting(5.3.37)and(5.3.38)into(5.3.33)and(5.3.34)yields
ADf
x
.X
0
/C
1
.t/; BDf
y
.X
0
/C
2
.t/;
(5.3.40)
where
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334 Chapter5
Real-ValuedFunctionsofSeveralVariables
1
.t/D
v
2
jUjE
1
.t/u
2
jVjE
2
.t/
u
1
v
2
u
2
v
1
and
2
.t/D
u
1
jVjE
2
.t/v
1
jUjE
1
.t/
u
1
v
2
u
2
v
1
;
so
lim
t!0
i
.t/D0; iD1;2;
(5.3.41)
becauseof(5.3.39).
From(5.3.29)and(5.3.40),theequationofthesecantplaneis
´Df.X
0
/CŒf
x
.X
0
/C
1
.t/.xx
0
/CŒf
y
.X
0
/C
2
.t/.yy
0
/:
Therefore,becauseof(5.3.41),thesecantplane“approaches”thetangentplane(5.3.28)as
tapproacheszero.
MaximaandMinima
WesaythatX
0
isalocalextremepointoff ifthereisaı>0suchthat
f.X/f.X
0
/
doesnotchangesigninS
ı
.X
0
/\D
f
.Morespecifically,X
0
isalocalmaximumpointif
f.X/f.X
0
/
oralocalminimumpointif
f.X/f.X
0
/
forallXinS
ı
.X
0
/\D
f
.
ThenexttheoremisanalogoustoTheorem2.3.7.
Theorem5.3.11
SupposethatfisdefinedinaneighborhoodofX
0
inR
n
andf
x
1
.X
0
/;
f
x
2
.X
0
/;...;f
x
n
.X
0
/exist:LetX
0
bealocalextremepointoff:Then
f
x
i
.X
0
/D0; 1in:
(5.3.42)
Proof
Let
E
1
D.1;0;:::;0/; E
2
D.0;1;0;:::;0/;:::; E
n
D.0;0;:::;1/;
and
g
i
.t/Df.X
0
CtE
i
/; 1in:
Theng
i
isdifferentiableattD0,with
g
0
i
.0/Df
x
i
.X
0
/
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Section5.3
PartialDerivativesandtheDifferential
335
(Definition5.3.1).SinceX
0
isalocalextremepointoff,t
0
D0isalocalextremepoint
ofg
i
.NowTheorem2.3.7impliesthatg
0
i
.0/D0,andthisimplies(5.3.42).
TheconverseofTheorem5.3.11isfalse,since(5.3.42)mayholdatapointX
0
thatis
notalocalextremepointoff.Forexample,letX
0
D.0;0/and
f.x;y/Dx
3
Cy
3
:
WesaythatapointX
0
where(5.3.42)holdsisacriticalpointoff. Thus,iff f isdefined
inaneighborhoodofalocalextremepointX
0
,thenX
0
isacriticalpointoff;however,a
criticalpointneednotbealocalextremepointoff.
TheuseofTheorem5.3.11forfindinglocalextremepointsiscoveredincalculus,sowe
willnotpursueithere.
5.3Exercises
1.
Calculate@f.X/=@ˆ.
(a)
f.x;y/Dx
2
C2xycosx, ˆD
1
p
3
;
r
2
3
!
(b)
f.x;y;´/De
xCy
2
C2´
, ˆD
1
p
3
;
1
p
3
;
1
p
3
(c)
f.X/DjXj
2
, ˆD
1
p
n
;
1
p
n
;;
1
p
n
(d)
f.x;y;´/Dlog.1CxCyC´/, ˆD.0;1;0/
2.
Let
f.x;y/D
8
<
:
xysinx
x2Cy2
; .x;y/¤.0;0/;
0;
.x;y/D.0;0/;
andletˆD.
1
;
2
/beaunitvector.Find@f.0;0/=@ˆ.
3.
Find@f.X
0
/=@ˆ,whereˆistheunitvectorinthedirectionofX
1
X
/
.
(a)
f.x;y;´/Dsinxy´; X
0
D.1;1;2/, X
1
D.3;2;1/
(b)
f.x;y;´/De
.x
2
Cy
2
C2´/
; X
0
D.1;0;1/, X
1
D.2;0;1/
(c)
f.x;y;´/Dlog.1CxCyC´/; X
0
D.1;0;1/, X
1
D.3;0;1/
(d)
f.X/DjXj4; X
0
D0, X
1
D.1;1;:::;1/
4.
Giveageometricalinterpretationofthedirectionalderivative@f.x
0
;y
0
/=@ˆofa
functionoftwovariables.
5.
Findallfirst-orderpartialderivatives.
(a)
f.x;y;´/Dlog.xCyC2´/
(b)
f.x;y;´/Dx
2
C3xy´C2xy
(c)
f.x;y;´/Dxe
(d)
f.x;y;´/D´Csinx
2
y
6.
Findallsecond-orderpartialderivativesofthefunctionsinExercise5.3.5.
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336 Chapter5
Real-ValuedFunctionsofSeveralVariables
7.
Findallsecond-orderpartialderivativesofthefollowingfunctionsat.0;0/.
(a)
f.x;y/D
8
<
:
xy.xy2
x2Cy2
; .x;y/¤.0;0/;
0;
.x;y/D.0;0/
(b)
f.x;y/D
(
x
2
tan
1
y
x
y
2
tan
1
x
y
; x¤0; y¤0;
0;
xD0 or yD0
(Herejtan
1
uj<=2.)
8.
Findafunctionf D D f.x;y/suchthatf
xy
existsforall.x;y/, butf
y
exists
nowhere.
9.
Letu andv be functionsoftwovariableswithcontinuoussecond-orderpartial
derivatives inaregionS. Supposethatu
x
D v
y
andu
y
D v
x
inS. Show
that
u
xx
Cu
yy
Dv
xx
Cv
yy
D0
inS.
10.
Letf beafunctionof.x
1
;x
2
;:::;x
n
/.n2/suchthatf
x
i
,f
x
j
,andf
x
i
x
j
.i¤
j/existonaneighborhoodofX
0
andf
x
i
x
j
iscontinuousatX
0
.UseTheorem5.3.3
toprovethatf
x
j
x
i
.X
0
/existsandequalsf
x
i
x
j
.X
0
/.
11.
UseExercise5.3.10andinductiononrtoproveTheorem5.3.4.
12.
Letr
1
;r
2
;:::;r
n
benonnegativeintegerssuchthat
r
1
Cr
2
CCr
n
Dr0:
(a)
Showthat
1
2
CC´
n
/
r
D
X
r
r
1
Šr
2
Šr
n
Š
´
r
1
1
´
r
2
2
´
r
n
n
;
where
P
r
denotes summationoveralln-tuples.r
1
;r
2
;:::;r
n
/thatsatisfy
thestatedconditions. H
INT
:Thisisobviousifn D D 1;anditfollowsfrom
Exercise1.2.19ifnD2:Useinductiononn:
(b)
Showthatthereare
r
1
Šr
2
Šr
n
Š
orderedn-tuplesofintegers.i
1
;i
2
;:::;i
n
/thatcontainr
1
ones,r
2
twos,...,
andr
n
n’s.
(c)
Letf beafunctionof.x
1
;x
2
;:::;x
n
/.Showthatthereare
r
1
Šr
2
Šr
n
Š
partialderivativesf
x
i
1
x
i
2
x
i
r
thatinvolvedifferentiationr
i
timeswithrespect
tox
i
,fori D1;2;:::;n.
13.
ProveLemma5.3.8.
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Section5.3
PartialDerivativesandtheDifferential
337
14.
Showthatthefunction
f.x;y/D
8
<
:
x
2
y
x6C2y2
; .x;y/¤.0;0/;
0;
.x;y/D.0;0/;
hasadirectionalderivativeinthedirectionofanarbitraryunitvectorˆat.0;0/,but
f isnotcontinuousat.0;0/.
15.
Prove:Iff
x
andf
y
areboundedinaneighborhoodof.x
0
;y
0
/,thenfiscontinuous
at.x
0
;y
0
/.
16.
ShowdirectlyfromDefinition5.3.5thatf isdifferentiableatX
0
.
(a)
f.x;y/D2x2C3xyCy2, X
0
D.1;2/
(b)
f.x;y;´/D2x
2
C3xC4y´, X
0
D.1;1;1/
(c)
f.X/DjXj
2
, X
0
arbitrary
17.
Supposethatf
x
existsonaneighborhoodof.x
0
;y
0
/andiscontinuousat.x
0
;y
0
/,
whilef
y
merelyexistsat.x
0
;y
0
/.Showthatf isdifferentiableat.x
0
;y
0
/.
18.
Finddfandd
X
0
f,andwrite.d
X
0
f/.XX
0
/.
(a)
f.x;y/Dx
3
C4xy
2
C2xysinx, X
0
D.0;2/
(b)
f.x;y;´/De
.xCyC´/
, X
0
D.0;0;0/
(c)
f.X/Dlog.1Cx
1
C2x
2
C3x
3
CCnx
n
/, X
0
D0
(d)
f.X/DjXj2r, X
0
D.1;1;1;:::;1/
19. (a)
Supposethatf isdifferentiableatX
0
andˆ D .
1
;
2
;:::;
n
/isaunit
vector.Showthat
@f.X
0
/
Df
x
1
.X
0
/
1
Cf
x
2
.X
0
/
2
CCf
x
n
.X
0
/
n
:
(b)
Forwhatunitvectorˆdoes@f.X
0
/=@ˆattainitsmaximumvalue?
20.
LetfbedefinedonRby
f.X/Dg.x
1
/Cg.x
2
/CCg.x
n
/;
where
g.u/D
(
u
2
sin
1
u
; u¤0;
0;
uD0:
Showthatf isdifferentiableat.0;0;:::;0/,butf
x
1
,f
x
2
,...,f
x
n
arealldiscon-
tinuousat.0;0;:::;0/.
21.
Thepurposeofthisexerciseistoshowthatiff,f
x
andf
y
existonaneighborhood
N of.x
0
;y
0
/ andf
x
andf
y
aredifferentiableat.x
0
;y
0
/, thenf
xy
.x
0
;y
0
/ D
f
yx
.x
0
;y
0
/.Supposethattheopensquare
˚
.x;y/
ˇ
ˇ
jxx
0
j<jhj;jyy
0
j<jhj
338 Chapter5
Real-ValuedFunctionsofSeveralVariables
isinN.Consider
B.h/Df.x
0
Ch;y
0
Ch/f.x
0
Ch;y
0
/f.x
0
;y
0
Ch/Cf.x
0
;y
0
/:
(a)
UsethemeanvaluetheoremaswedidintheproofofTheorem5.3.3towrite
B.h/DŒf
x
.bx;y
0
Ck/f
x
.bx;y
0
/h;
wherebxisbetweenx
0
andx
0
Ch. Thenusethedifferentiabilityoff
x
at
.x
0
;y
0
/toinferthat
B.h/Dh
2
f
xy
.x
0
;y
0
/ChE
1
.h/; where
lim
h!0
E
1
.h/
h
D0:
(b)
Usethemeanvaluetheoremtowrite
B.h/D
f
y
.x
0
Ch;by/f
y
.x
0
;by/
h;
wherebyisbetweeny
0
andy
0
Ch. Thenusethedifferentiabilityoff
y
at
.x
0
;y
0
/toinferthat
B.h/Dh
2
f
yx
.x
0
;y
0
/ChE
2
.h/; where
lim
h!0
E
2
.h/
h
D0:
(c)
Inferfrom
(a)
and
(b)
thatf
xy
.x
0
;y
0
/Df
yx
.x
0
;y
0
/.
22. (a)
Letf
x
i
andf
x
j
be differentiableatapointX
0
inR
n
. ShowfromExer-
cise5.3.21that
f
x
i
x
j
.X
0
/Df
x
j
x
i
.X
0
/:
(b)
Use
(a)
andinductiononrtoshowthatall.r1/-storderpartialderivatives
offaredifferentiableonanopensubsetSofRn,thenf
x
i
1
x
i
2
x
i
r
.X/(X2S)
dependsonlyonthenumberofdifferentiationswithrespecttoeachvariable,
andnotontheorderinwhichtheyareperformed.
23.
Provethat.x
0
;y
0
/,.x
1
;y
1
/,and.x
2
;y
2
/lieonalineifandonlyif
.x
1
x
0
/.y
2
y
0
/.x
2
x
0
/.y
1
y
0
/D0:
24.
Findtheequationofthetangentplanetothesurface
´Df.x;y/ at .x
0
;y
0
0
/D.x
0
;y
0
;f.x
0
;y
0
//:
(a)
f.x;y/Dx
2
Cy
2
1; .x
0
;y
0
/D.1;2/
(b)
f.x;y/D2xC3yC1; .x
0
;y
0
/D.1;1/
(c)
f.x;y/Dxysinxy; .x
0
;y
0
/D.1;=2/
(d)
f.x;y/Dx22yC3xy; .x
0
;y
0
/D.2;1/
Section5.4
TheChainRuleandTaylor’sTheorem
339
25.
Prove:Iff isdifferentiableat.x
0
;y
0
/and
lim
.x;y/!.x
0
;y
0
/
f.x;y/ab.xx
0
/c.yy
0
/
p
.xx
0
/2C.yy
0
/2
D0;
thenaDf.x
0
;y
0
/,bDf
x
.x
0
;y
0
/,andcDf
y
.x
0
;y
0
/.
5.4THECHAINRULEANDTAYLOR’STHEOREM
Wenowconsidertheproblemofdifferentiatingacompositefunction
h.U/Df.G.U//;
whereG D D .g
1
;g
2
;:::;g
n
/isavector-valuedfunction,asdefinedinSection5.2. We
beginwiththefollowingdefinition.
Definition5.4.1
Avector-valuedfunctionGD.g
1
;g
2
;:::;g
n
/isdifferentiableat
U
0
D.u
10
;u
20
;:::;u
m0
/
ifitscomponentfunctionsg
1
,g
2
,...,g
n
aredifferentiableatU
0
.
Weneedthefollowinglemmatoprovethemainresultofthesection.
Lemma5.4.2
SupposethatGD.g
1
;g
2
;:::;g
n
/isdifferentiableat
U
0
D.u
10
;u
20
;:::;u
m0
/;
anddefine
MD
0
@
Xn
iD1
Xm
jD1
@g
i
.U
0
@u
j
2
1
A
1=2
:
Then;if>0;thereisaı>0suchthat
jG.U/G.U
0
/j
jUU
0
j
<MC if 0<jUU
0
j<ı:
Proof
Sinceg
1
,g
2
,...,g
n
aredifferentiableatU
0
,applyingLemma5.3.8tog
i
shows
that
g
i
.U/g
i
.U
0
/D.d
U
0
g
i
/.UU
0
/CE
i
.U/j.UU
0
j
D
Xm
jD1
@g
i
.U
0
/
@u
j
.u
j
u
j0
/CE
i
.U/j.UU
0
j;
(5.4.1)
340 Chapter5
Real-ValuedFunctionsofSeveralVariables
where
lim
U!U
0
E
i
.U/D0; 1in:
(5.4.2)
FromSchwarz’sinequality,
jg
i
.U/g
i
.U
0
/j.M
i
CjE
i
.U/j/jUU
0
j;
where
M
i
D
0
@
Xm
jD1
@g
i
.U
0
/
@u
j
2
1
A
1=2
:
Therefore,
jG.U/G.U
0
/j
jUU
0
j
Xn
iD1
.M
i
CjE
i
.U/j/
2
!
1=2
:
From(5.4.2),
lim
U!U
0
Xn
iD1
.M
i
CjE
i
.U/j/
2
!
1=2
D
Xn
iD1
M
2
i
!
1=2
DM;
whichimpliestheconclusion.
ThefollowingtheoremisanalogoustoTheorem2.3.5.
Theorem5.4.3(TheChainRule)
Supposethatthereal-valuedfunctionf is
differentiableatX
0
inR
n
;thevector-valuedfunctionGD.g
1
;g
2
;:::;g
n
/isdifferentiable
atU
0
inR
m
;andX
0
DG.U
0
/:Thenthereal-valuedcompositefunctionhDfıGdefined
by
h.U/Df.G.U//
(5.4.3)
isdifferentiableatU
0
;and
d
U
0
hDf
x
1
.X
0
/d
U
0
g
1
Cf
x
2
.X
0
/d
U
0
g
2
CCf
x
n
.X
0
/d
U
0
g
n
:
(5.4.4)
Proof
WeleaveittoyoutoshowthatU
0
isaninteriorpointofthedomainofh(Exer-
cise5.4.1),soitislegitimatetoaskifhisdifferentiableatU
0
.
LetX
0
D.x
10
;x
20
;:::;x
n0
/.Notethat
x
i0
Dg
i
.U
0
/; 1i i n;
byassumption.Sincef isdifferentiableatX
0
,Lemma5.3.8impliesthat
f.X/f.X
0
/D
Xn
iD1
f
x
i
.X
0
/.x
i
x
i0
/CE.X/jXX
0
j;
(5.4.5)
where
lim
X!X
0
E.X/D0:
Section5.4
TheChainRuleandTaylor’sTheorem
341
SubstitutingXDG.U/andX
0
DG.U
0
/in(5.4.5)andrecalling(5.4.3)yields
h.U/h.U
0
/D
Xn
iD1
f
x
i
.X
0
/.g
i
.U/g
i
.U
0
//CE.G.U//jG.U/G.U
0
/j: (5.4.6)
Substituting(5.4.1)into(5.4.6)yields
h.U/h.U
0
/D
Xn
iD1
f
x
i
.X
0
/.d
U
0
g
i
/.UU
0
/C
Xn
iD1
f
x
i
.X
0
/E
i
.U/
!
jUU
0
j
CE.G.U//jG.U/G.U
0
j:
Since
lim
U!U
0
E.G.U//D lim
X!X
0
E.X/D0;
(5.4.2)andLemma5.4.2implythat
h.U/h.U
0
/
Xn
iD1
f
x
i
.X
0
d
U
0
g
i
.UU
0
/
jUU
0
j
D0:
Therefore,hisdifferentiableatU
0
,andd
U
0
hisgivenby(5.4.4).
Example5.4.1
Let
f.x;y;´/D2x
2
C4xyC3y´;
g
1
.u;v/Du
2
Cv
2
; g
2
.u;v/Du
2
2v
2
; g
3
.u;v/Duv;
and
h.u;v/Df.g
1
.u;v/;g
2
.u;v/;g
3
.u;v//:
LetU
0
D.1;1/and
X
0
D.g
1
.U
0
/;g
2
.U
0
/;g
3
.U
0
//D.2;1;1/:
Then
f
x
.X
0
/D4; f
y
.X
0
/D5; f
´
.X
0
/D3;
@g
1
.U
0
/
@u
D
2;
@g
1
.U
0
/
@v
D2;
@g
2
.U
0
/
@u
D
2;
@g
2
.U
0
/
@v
D
4;
@g
3
.U
0
/
@u
D1;
@g
3
.U
0
/
@v
D
1:
Therefore,
d
U
0
g
1
D2du2dv; d
U
0
g
2
D2duC4dv; d
U
0
g
3
DduCdv;
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