c# pdf free : Chrome pdf from link Library control component asp.net web page winforms mvc TRENCH_REAL_ANALYSIS36-part255

352 Chapter5
Real-ValuedFunctionsofSeveralVariables
Theorem5.4.9
Supposethatfanditspartialderivativesoforderk1arediffer-
entiableinaneighborhoodN ofapointX
0
inR
n
andallkth-orderpartialderivativesof
f arecontinuousatX
0
:Then
lim
X!X
0
f.X/T
k
.X/
jXX
0
jk
D0:
(5.4.31)
Proof
If > 0,thereisaı > 0suchthatB
ı
.X
0
/  N andallkth-orderpartial
derivativesoff satisfytheinequality
ˇ
ˇ
ˇ
ˇ
ˇ
@
k
f.
e
X/
@x
i
k
@x
i
k1
@x
i
1
@
k
f.X
0
/
@x
i
k
@x
i
k1
@x
i
1
ˇ
ˇ
ˇ
ˇ
ˇ
<;
e
X2B
ı
.X
0
/:
(5.4.32)
NowsupposethatX2B
ı
.X
0
/.FromTheorem5.4.8withkreplacedbyk1,
f.X/DT
k1
.X/C
1
.d
.k/
e
X
f/.XX
0
/;
(5.4.33)
where
e
XissomepointonthelinesegmentfromX
0
toXandisthereforeinB
ı
.X
0
/. We
canrewrite(5.4.33)as
f.X/DT
k
.X/C
1
h
.d
.k/
e
X
f/.XX
0
/.d
.k/
X
0
f/.XX
0
/
i
:
(5.4.34)
But(5.4.23)and(5.4.32)implythat
ˇ
ˇ
ˇ
.d
.k/
e
X
f/.XX
0
/.d
.k/
X
0
f/.XX
0
/
ˇ
ˇ
ˇ
<n
k
jXX
0
j
k
(5.4.35)
(Exercise5.4.17),whichimpliesthat
jf.X/T
k
.X/j
jXX
0
jk
<
n
k
; X2B
ı
.X
0
/;
from(5.4.34).Thisimplies(5.4.31).
LetrbeapositiveintegerandX
0
D.x
10
;x
20
;:::;x
n0
/.Afunctionoftheform
p.X/D
X
r
a
r
1
r
2
:::r
n
.x
1
x
10
/
r
1
.x
2
x
20
/
r
2
.x
n
x
n0
/
r
n
;
(5.4.36)
wherethecoefficientsfa
r
1
r
2
:::r
n
gareconstantsandthesummationisoveralln-tuplesof
nonnegativeintegers.r
1
;r
2
;:::;r
n
/suchthat
r
1
Cr
2
CCr
n
Dr;
isahomogeneouspolynomialofdegreer inXX
0
, providedthatatleastoneofthe
coefficientsisnonzero.Forexample,iff satisfiestheconditionsofDefinition5.4.7,then
thefunction
p.X/D.d
.r/
X
0
f/.XX
0
/
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Section5.4
TheChainRuleandTaylor’sTheorem
353
issuchapolynomialifatleastoneoftherth-ordermixedpartialderivativesoff atX
0
is
nonzero.
Clearly,p.X
0
/ D D 0ifpisahomogeneouspolynomialofdegreer  1inXX
0
.
Ifp.X/  0forallX,wesaythatpispositivesemidefinite;ifp.X/ >0exceptwhen
XDX
0
,pispositivedefinite.
Similarly,pisnegativesemidefiniteifp.X/0ornegativedefiniteifp.X/<0forall
X¤X
0
.Inallthesecases,pissemidefinite.
Withpasin(5.4.36),
p.XC2X
0
/D.1/
r
p.X/;
sopcannotbesemidefiniteifrisodd.
Example5.4.10
Thepolynomial
p.x;y;´/Dx
2
Cy
2
2
CxyCx´Cy´
ishomogeneousofdegree2inXD.x;y;´/.Wecanrewritepas
p.x;y;´/D
1
2
.xCy/
2
C.yC´/
2
C.´Cx/
2
;
sopisnonnegative,andp.
x;
y;
´/D0ifandonlyif
xC
yD
yC
´D
´C
xD0;
whichisequivalentto.
x;
y;
´/ D .0;0;0/. Therefore, , pispositivedefiniteandpis
negativedefinite.
Thepolynomial
p
1
.x;y;´/Dx
2
Cy
2
2
C2xy
canberewrittenas
p
1
.x;y;´/D.xCy/
2
2
;
sop
1
isnonnegative. Sincep
1
.1;1;0/ D D 0, , p
1
ispositivesemidefiniteandp
1
is
negativesemidefinite.
Thepolynomial
p
2
.x;y;´/Dx
2
y
2
2
isnotsemidefinite,since,forexample,
p
2
.1;0;0/D1 and p
2
.0;1;0/D1:
FromTheorem5.3.11,iff isdifferentiableandattainsalocalextremevalueatX
0
,then
d
X
0
f D0;
(5.4.37)
sincef
x
1
.X
0
/Df
x
2
.X
0
/DDf
x
n
.X
0
/D0.However,theconverseisfalse.Thenext
theoremprovidesamethodfordecidingwhetherapointsatisfying(5.4.37)isanextreme
point.ItisrelatedtoTheorem2.5.3.
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354 Chapter5
Real-ValuedFunctionsofSeveralVariables
Theorem5.4.10
Supposethatf satisfiesthehypothesesofTheorem5.4.9withk
2;and
d
.r/
X
0
f 0 .1rk1/; d
.k/
X
0
f 60:
(5.4.38)
Then
(a)
X
0
isnotalocalextremepointoff unlessd
.k/
X
0
f issemidefiniteasapolynomialin
XX
0
:Inparticular;X
0
isnotalocalextremepointoff ifkisodd:
(b)
X
0
isalocalminimumpointoff ifd
.k/
X
0
f ispositivedefinite;oralocalmaximum
pointifd
.k/
X
0
f isnegativedefinite:
(c)
Ifd
.k/
X
0
f issemidefinite;thenX
0
maybealocalextremepointoff;butitneednot
be:
Proof
From(5.4.38)andTheorem5.4.9,
lim
X!X
0
f.X/f.X
0
/
1
.d
.k/
X
0
/.XX
0
/
jXX
0
jk
D0:
(5.4.39)
IfXDX
0
CtU,whereUisaconstantvector,then
.d
.k/
X
0
f/.XX
0
/Dt
k
.d
.k/
X
0
f/.U/;
so(5.4.39)impliesthat
lim
t!0
f.X
0
CtU/f.X
0
/
t
k
.d
.k/
X
0
f/.U/
tk
D0;
or,equivalently,
lim
t!0
f.X
0
CtU/f.X
0
/
tk
D
1
.d
.k/
X
0
f/.U/
(5.4.40)
foranyconstantvectorU.
Toprove
(a)
,supposethatd
.k/
X
0
f isnotsemidefinite.ThentherearevectorsU
1
andU
2
suchthat
.d
.k/
X
0
f/.U
1
/>0 and .d
.k/
X
0
f/.U
2
/<0:
Thisand(5.4.40)implythat
f.X
0
CtU
1
/>f.X
0
/ and f.X
0
CtU
2
/<f.X
0
/
fortsufficientlysmall.Hence,X
0
isnotalocalextremepointoff.
Toprove
(b)
,firstassumethatd
.k/
X
0
f ispositivedefinite. Thenitcanbeshownthat
thereisa>0suchthat
.d
.k/
X
0
f/.XX
0
/
jXX
0
j
k
(5.4.41)
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Section5.4
TheChainRuleandTaylor’sTheorem
355
forallX(Exercise5.4.19).From(5.4.39),thereisaı>0suchthat
f.X/f.X
0
/
1
.d
.k/
X
0
f/.XX
0
/
jXX
0
jk
>
2
if jXX
0
j<ı:
Therefore,
f.X/f.X
0
/>
1
.d
.k/
X
0
/.XX
0
/
2
jXX
0
j
k
if jXX
0
j<ı:
Thisand(5.4.41)implythat
f.X/f.X
0
/>
2
jXX
0
j
k
if jXX
0
j<ı;
whichimpliesthatX
0
isalocalminimumpointoff. Thisproveshalfof
(b)
. Weleave
theotherhalftoyou(Exercise5.4.20).
Toprove
(c)
merelyrequiresexamples;seeExercise5.4.21.
Corollary5.4.11
Supposethatf;f
x
;andf
y
aredifferentiableinaneigborhoodofa
criticalpointX
0
D.x
0
;y
0
/off andf
xx
;f
yy
;andf
xy
arecontinuousat.x
0
;y
0
/:Let
DDf
xx
.x
0
;y
0
/f
xy
.x
0
;y
0
/f
2
xy
.x
0
;y
0
/:
Then
(a)
.x
0
;y
0
/isalocalextremepointoff ifD>0I.x
0
;y
0
/isalocalminimumpointif
f
xx
.x
0
;y
0
/>0,oralocalmaximumpointiff
xx
.x
0
;y
0
/<0:
(b)
.x
0
;y
0
/isnotalocalextremepointoff ifD<0:
Proof
Write.xx
0
;yy
0
/D.u;v/and
p.u;v/D.d
.2/
X
0
f/.u;v/DAu
2
C2BuvCCv
2
;
whereADf
xx
.x
0
;y
0
/,BDf
xy
.x
0
;y
0
/,andC Df
yy
.x
0
;y
0
/,so
DDAC B
2
:
IfD>0,thenA¤0,andwecanwrite
p.u;v/DA
u
2
C
2B
A
uvC
B
2
A2
v
2
C
C
B
2
A
v
2
DA
uC
B
A
v
2
C
D
A
v
2
:
Thiscannotvanishunlessu D v D 0. Hence,d
.2/
X
0
f ispositivedefiniteifA A > > 0or
negativedefiniteifA<0,andTheorem5.4.10
(b)
implies
(a)
.
IfD<0,therearethreepossibilities:
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356 Chapter5
Real-ValuedFunctionsofSeveralVariables
1.
A¤0;thenp.1;0/DAandp
B
A
;1
D
D
A
.
2.
C¤0;thenp.0;1/DC andp
1;
B
C
D
D
C
.
3.
ADC D0;thenB¤0andp.1;1/D2Bandp.1;1/D2B.
Ineachcasethetwogivenvaluesofpdifferinsign,soX
0
isnotalocalextremepoint
off,fromTheorem5.4.10
(a)
.
Example5.4.11
If
f.x;y/De
ax
2
Cby
2
;
then
f
x
.x;y/D2axf.x;y/; f
y
.x;y/D2byf.x;y/;
so
f
x
.0;0/Df
y
.0;0/D0;
and.0;0/isacriticalpointoff.ToapplyCorollary5.4.11,wecalculate
f
xx
.x;y/D.2aC4a
2
x
2
/f.x;y/;
f
yy
.x;y/D.2bC4b
2
y
2
/f.x;y/;
f
xy
.x;y/D4abxyf.x;y/:
Therefore,
DDf
xx
.0;0/f
yy
.0;0/f
2
xy
.0;0/D.2a/.2b/.0/.0/D4ab:
Corollary5.4.11impliesthat.0;0/isalocalminimumpointifaandbarepositive,alocal
maximumifaandbarenegative,andneitherifoneispositiveandtheotherisnegative.
Corollary5.4.11doesnotapplyifaorbiszero.
5.4Exercises
Intheexercisesontheuseofthechainrule,assumethatthefunctionssatisfyappropriate
differentiabilityconditions.
1.
UndertheassumptionsofTheorem5.4.3,showthatU
0
isaninteriorpointofthe
domainofh.
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Section5.4
TheChainRuleandTaylor’sTheorem
357
2.
Leth.U/ D f.G.U// / andfindd
U
0
hbyTheorem5.4.3, andthenbywritingh
explicitlyasafunctionofU.
(a)
f.x;y/ D D 3x
2
C4xy
2
C3x,
g
1
.u;v/ D D ve
uCv1
,
g
2
.u;v/ D D e
uCv1
,
.u
0
;v
0
/D.0;1/
(b)
f.x;y;´/D e.xCyC´/,
g
1
.u;v;w/D logulogvClogw,
g
2
.u;v;w/D 2logu3logw,
g
3
.u;v;w/D loguClogvC2logw,
.u
0
;v
0
;w
0
/D.1;1;1/
(c)
f.x;y/ D .xCy/
2
,
g
1
.u;v/ D ucosv,
g
2
.u;v/ D usinv,
.u
0
;v
0
/D.3;=2/
(d)
f.x;y;´/ D x
2
Cy
2
2
,
g
1
.u;v;w/ D ucosvsinw,
g
2
.u;v;w/ D ucosvcosw,
g
3
.u;v;w/ D usinv;
.u
0
;v
0
;w
0
/D.4;=3;=6/
3.
Leth.r;;´/Df.x;y;´/,wherex DrcosandyDrsin. . Findh
r
,h
,and
h
´
intermsoff
x
,f
y
,andf
´
.
4.
Leth.r;;/ Df.x;y;´/,wherex D rsincos,y y D D rsinsin,and´ D
rcos.Findh
r
,h
,andh
intermsoff
x
,f
y
,andf
´
.
5.
Prove:
(a)
Ifh.u;v/Df.u2Cv2/,thenvh
u
uh
v
D0.
(b)
Ifh.u;v/Df.sinuCcosv/,thenh
u
sinvCh
v
cosuD0.
(c)
Ifh.u;v/Df.u=v/,thenuh
u
Cvh
v
D0.
(d)
Ifh.u;v/Df.g.u;v/;g.u;v//,thendhD.f
x
f
y
/dg.
6.
Findh
y
andh
´
if
h.y;´/Dg.x.y;´/;y;´;w.y;´//:
7.
Supposethatu,v,andf aredefinedon.1;1/. . Letuandvbedifferentiable
andf becontinuousforallx.Showthat
d
dx
Z
v.x/
u.x/
f.t/dt Df.v.x//v
0
.x/f.u.x//u
0
.x/:
8.
Wesaythatf Df.x
1
;x
2
;:::;x
n
/ishomogeneousofdegreerifD
f
isopenand
thereisaconstantrsuchthat
f.tx
1
;tx
2
;:::;tx
n
/Dt
r
f.x
1
;x
2
;:::;x
n
/
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358 Chapter5
Real-ValuedFunctionsofSeveralVariables
whenevert >0and.x
1
;x
2
;:::;x
n
/and.tx
1
;tx
2
;:::;tx
n
/areinD
f
. Prove:If
f isdifferentiableandhomogeneousofdegreer,then
Xn
iD1
x
i
f
x
i
.x
1
;x
2
;:::;x
n
/Drf.x
1
;x
2
;:::;x
n
/:
(ThisisEuler’stheoremforhomogeneousfunctions.)
9.
Ifh.r;/Df.rcos;rsin/,showthat
f
xx
Cf
yy
Dh
rr
C
1
r
h
r
C
1
r2
h

:
H
INT
:Rewritethedefiningequationasf.x;y/Dh.r.x;y/;.x;y//;withr.x;y/D
p
x2Cy2and.x;y/Dtan
1
.y=x/;anddifferentiatewithrespecttoxandy:
10.
Leth.u;v/Df.a.u;v/;b.u;v//,wherea
u
Db
v
anda
v
Db
u
.Showthat
h
uu
Ch
vv
D.f
xx
Cf
yy
/.a
2
u
Ca
2
v
/:
11.
Prove:If
u.x;t/Df.xct/Cg.xCct/;
thenu
tt
Dc
2
u
xx
.
12.
Leth.u;v/Df.uCv;uv/.Showthat
(a)
f
xx
f
yy
Dh
uv
(b)
f
xx
Cf
yy
D
1
2
.h
uu
Ch
vv
/
13.
ReturningtoExercise5.4.4,findh
rr
andh
r
intermsofthepartialderivativesof
f.
14.
Leth
uv
D0forall.u;v/.Showthathisoftheform
h.u;v/DU.u/CV.v/:
UsethisandExercise5.4.12
(a)
toshowthatiff
xx
f
yy
D0forall.x;y/,then
f.x;y/DU.xCy/CV.xy/:
15.
Proveorgiveacounterexample: Iff isdifferentiableandf
x
D0inaregionD,
thenf.x
1
;y/D f.x
2
;y/whenever.x
1
;y/and.x
2
;y/areinD;thatisf.x;y/
dependsonlyony.
16.
FindT
3
.X/.
(a)
f.x;y/De
x
cosy, X
0
D.0;0/
(b)
f.x;y/De
xy
, X
0
D.0;0/
(c)
f.x;y;´/D.xCyC´3/
5
, X
0
D.1;1;1/
(d)
f.x;y;´/Dsinxsinysin´, X
0
D.0;0;0/
17.
UseEqns.(5.4.23)and(5.4.32)toproveEqn.(5.4.35).
Section5.4
TheChainRuleandTaylor’sTheorem
359
18.
CarefullyexplainwhyTheorem5.4.9isnotageneralizationofTheorem2.5.1.
19.
SupposethatpisahomogeneouspolynomialofdegreerinYandp.Y/>0forall
nonzeroYinR
n
. Showthatthereisa>0suchthatp.Y/jYj
r
forallYin
Rn.H
INT
:passumesaminimumontheset
˚
Y
ˇ
ˇ
jYjD1
:Usethistoestablishthe
inequalityinEqn.(5.4.41):
20.
CompletetheproofofTheorem5.4.10
(b)
.
21. (a)
Showthat.0;0/isacriticalpointofeachofthefollowingfunctions,andthat
theyhavepositivesemidefiniteseconddifferentialsat.0;0/.
p.x;y/Dx
2
2xyCy
2
Cx
4
Cy
4
I
q.x;y/Dx
2
2xyCy
2
x
4
y
4
:
(b)
ShowthatDasdefinedinCorollary5.4.11iszeroforbothpandq.
(c)
Showthat.0;0/isalocalminimumpointofpbutnotalocalextremepoint
ofq.
22.
Supposethatp D D p.x
1
;x
2
;:::;x
n
/isahomogeneouspolynomialofdegreer
(Exercise5.4.8).Leti
1
,i
2
,...,i
n
benonnegativeintegerssuchthat
i
1
Ci
2
CCi
n
Dk;
andlet
q.x
1
;x
2
;:::;x
n
/D
@
k
p.x
1
;x
2
;:::;x
n
/
@x
i
1
1
@x
i
2
2
@x
i
n
n
:
Showthatqishomogeneousofdegree rk,subjecttotheconventionthata
homogeneouspolynomialofnegativedegreeisidenticallyzero.
23.
Supposethatf D D f.x
1
;x
2
;:::;x
n
/isahomogeousfunctionofdegreer(Exer-
cise8),withmixedpartialderivativeofallorders.Showthat
Xn
i;jD1
x
i
x
j
@2f.x
1
;x
2
;:::;x
n
/
@x
i
@x
j
Dr.r1/f.x
1
;x
2
;:::;x
n
/
and
Xn
i;j;kD1
x
i
x
j
x
k
@
3
.x
1
;x
2
;:::;x
n
/
@x
i
@x
j
@x
k
Dr.r1/.r2/f.x
1
;x
2
;:::;x
n
/:
Canyougeneralizetheseresults?
24.
ObtaintheresultinExample5.4.7bywriting
F.X/De
a
1
x
1
e
a
2
x
2
e
a
n
x
n
;
formallymultiplyingtheseries
e
a
i
x
i
D
X1
r
i
D0
.1/
r
i
.a
i
x
i
/
r
i
r
i
Š
; 1in
together,andcollectingtheresultingproductsappropriately.
360 Chapter5
Real-ValuedFunctionsofSeveralVariables
25.
Let
f.x;y/De
xCy
:
Bywriting
f.x;y/D
1
X
rD0
.xCy/
r
;
andexpanding.xCy/
r
bymeansofthebinomialtheorem,verifythat
d
.r/
(
0;0/
f D
Xr
jD0
r
j
!
@
r
f.0;0/
@xj@yrj
.dx/
j
.dy/
rj
:
CHAPTER6
Vector-ValuedFunctions
ofSeveralVariables
INTHISCHAPTERwestudythedifferentialcalculusofvector-valuedfunctionsofseveral
variables.
SECTION6.1reviewsmatrices,determinants,andlineartransformations,whichareinte-
gralpartsofthedifferentialcalculusaspresentedhere.
SECTION6.2definescontinuityanddifferentiabilityofvector-valuedfunctionsofseveral
variables.Thedifferentialofavector-valuedfunctionFisdefinedasacertainlineartrans-
formation. ThematrixofthislineartransformationiscalledthedifferentialmatrixofF,
denotedbyF
0
. Thechainruleisextendedtocompositionsofdifferentiablevector-valued
functions.
SECTION6.3presentsacompleteproofoftheinversefunctiontheorem.
SECTION6.4.usestheinversefunctiontheoremtoprovetheimplicitfunctiontheorem.
6.1LINEARTRANSFORMATIONSANDMATRICES
Inthisandsubsequentsectionsitwilloftenbeconvenienttowritevectorsvertically;thus,
insteadofXD.x
1
;x
2
;:::;x
n
/wewillwrite
XD
2
6
6
6
4
x
1
x
2
:
:
:
x
n
3
7
7
7
5
whendealingwithmatrixoperations. Althoughweassumethatyouhavecompleteda
courseinlinearalgebra,wewillreviewthepertinentmatrixoperations.
Wehavedefinedvector-valuedfunctionsasorderedn-tuplesofreal-valuedfunctions,in
connectionwithcompositefunctionshDf ıG,wheref isreal-valuedandGisvector-
valued.Wenowconsidervector-valuedfunctionsasobjectsofinterestontheirown.
361
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