342 Chapter5
Real-ValuedFunctionsofSeveralVariables
and,from(5.4.4),
d
U
0
hDf
x
.X
0
/d
U
0
g
1
Cf
y
.X
0
/d
U
0
g
2
Cf
´
.X
0
/d
U
0
g
3
D4.2du2dv/C5.2duC4dv/3.duCdv/
D21duC9dv:
Since
d
U
0
hDh
u
.U
0
/duCh
v
.U
0
/dv
weconcludethat
h
u
.U
0
/D21 and h
v
.U
0
/D9:
(5.4.7)
Thiscanalsobeobtainedbywritinghexplicitlyintermsof.u;v/anddifferentiating;thus,
h.u;v/D2Œg
1
.u;v/
2
C4g
1
.u;v/g
2
.u;v/C3g
2
.u;v/g
3
.u;v/
D2.u
2
Cv
2
/
2
C4.u
2
Cv
2
/.u
2
2v
2
/C3.u
2
2v
2
/uv
D6u
4
C3u
3
v6uv
3
6v
4
:
Hence,
h
u
.u;v/D24u
3
C9u
2
v6v
3
and
h
v
.u;v/D3u
3
18uv
2
24v
3
;
soh
u
.1;1/D21andh
v
.1;1/D9,consistentwith(5.4.7).
Corollary5.4.4
UndertheassumptionsofTheorem5.4.3;
@h.U
0
/
@u
i
D
Xn
jD1
@f.X
0
/
@x
j
@g
j
.U
0
/
@u
i
; 1im:
(5.4.8)
Proof
Substituting
d
U
0
g
i
D
@g
i
.U
0
/
@u
1
du
1
C
@g
i
.U
0
/
@u
2
du
2
CC
@g
i
.U
0
/
@u
m
du
m
; 1in;
into(5.4.4)andcollectingmultipliersofdu
1
,du
2
,...,du
m
yields
d
U
0
hD
m
X
iD1
0
@
n
X
jD1
@f.X
0
/
@x
j
@g
j
.U
0
/
@u
i
1
A
du
i
:
However,fromTheorem5.3.6,
d
U
0
hD
m
X
iD1
@h.U
0
/
@u
i
du
i
:
Comparingthelasttwoequationsyields(5.4.8).
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Section5.4
TheChainRuleandTaylor’sTheorem
343
WhenitisnotimportanttoemphasizetheparticularpointX
0
,wewrite(5.4.8)less
formallyas
@h
@u
i
D
n
X
jD1
@f
@x
j
@g
j
@u
i
; 1i i m;
(5.4.9)
withtheunderstandingthatincalculating@h.U
0
/=@u
i
,@g
j
=@u
i
isevaluatedatU
0
and
@f=@x
j
atX
0
DG.U
0
/.
Theformulas(5.4.8)and(5.4.9)canalsobesimplifiedbyreplacingthesymbolGwith
XDX.U/;thenwewrite
h.U/Df.X.U//
and
@h.U
0
/
@u
i
D
Xn
jD1
@f.X
0
/
@x
j
@x
j
.U
0
/
@u
i
;
orsimply
@h
@u
i
D
Xn
jD1
@f
@x
j
@x
j
@u
i
:
(5.4.10)
Example5.4.2
Let.r;/bepolarcoordinatesinthexy-plane;thatis,
xDrcos; yDrsin:
Supposethatf Df.x;y/isdifferentiableonasetS,andlet
h.r;/Df.rcos;rsin/:
If.rcos;rsin/2S,(5.4.10)impliesthat
@h
@r
D
@f
@x
@x
@r
C
@f
@y
@y
@r
Dcos
@f
@x
Csin
@f
@y
(5.4.11)
and
@h
@
D
@f
@x
@x
@
C
@f
@y
@y
@
Drsin
@f
@x
Crcos
@f
@y
;
wheref
x
andf
y
areevaluatedat.x;y/D.rcos;rsin/.
TheproofofCorollary5.4.4suggestsa straightforwardwaytocalculatethepartial
derivativesofacompositefunctionwithoutusing(5.4.10)explicitly.Ifh.U/Df.X.U//,
thenTheorem5.4.3,inthemorecasualnotationintroducedbeforeExample5.4.2,implies
that
dhDf
x
1
dx
1
Cf
x
2
dx
2
CCf
x
n
dx
n
;
(5.4.12)
wheredx
1
,dx
2
,...,dx
n
mustbewrittenintermsofthedifferentialsdu
1
,du
2
,...,du
m
oftheindependentvariables;thus,
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344 Chapter5
Real-ValuedFunctionsofSeveralVariables
dx
i
D
@x
i
@u
1
du
1
C
@x
i
@u
2
du
2
CC
@x
i
@u
m
du
m
:
Substitutingthisinto(5.4.12)andcollectingthemultipliersofdu
1
,du
2
,...,du
m
yields(5.4.10).
Example5.4.3
If
h.r;;´/Df.x.r;/;y.r;/;´/;
then
dhDf
x
dxCf
y
dyCf
´
d´:
But
dxD
@x
@r
drC
@x
@
d and dyD
@y
@r
drC
@y
@
dI
hence,
dhDf
x
@x
@r
drC
@x
@
d
Cf
y
@y
@r
drC
@y
@
d
Cf
´
D
f
x
@x
@r
Cf
y
@y
@r
drC
f
x
@x
@
Cf
y
@y
@
dCf
´
d´;
so
h
r
Df
x
@x
@r
Cf
y
@y
@r
; h
Df
x
@x
@
Cf
y
@y
@
; h
´
Df
´
:
Example5.4.4
Let
h.x/Df.x;y.x;´.x//;´.x//:
Then
dhDf
x
dxCf
y
dyCf
´
d´;
(5.4.13)
dyDy
x
dxCy
´
d´;
(5.4.14)
and
d´D´
0
dx;
(5.4.15)
wheretheprime indicatesdifferentiationwithrespecttox. Substituting(5.4.15)into
(5.4.14)yields
dyD.y
x
Cy
´
´
0
/dx
andsubstitutingthisand(5.4.15)into(5.4.13)yields
dhDŒf
x
Cf
y
.y
x
Cy
´
´
0
/Cf
´
´
0
dxI
hence,
h
0
Df
x
Cf
y
.y
x
Cy
´
´
0
/Cf
´
´
0
:
Heref
x
, f
y
, andf
´
areevaluatedat.x;y.x;´.x//;´.x//, y
x
andy
´
areevaluatedat
.x;´.x//,and´
0
isevaluatedatx.
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Section5.4
TheChainRuleandTaylor’sTheorem
345
HigherDerivativesofComposite Functions
Higherderivativesofcompositefunctionscanbecomputedbyrepeatedlyapplyingthe
chainrule.Forexample,differentiating(5.4.10)withrespecttou
k
yields
@
2
h
@u
k
@u
i
D
Xn
jD1
@
@u
k
@f
@x
j
@x
j
@u
i
D
Xn
jD1
@f
@x
j
@
2
x
j
@u
k
@u
i
C
Xn
jD1
@x
j
@u
i
@
@u
k
@f
@x
j
:
(5.4.16)
Wemustbecarefulfinding
@
@u
k
@f
@x
j
;
whichreallystandsherefor
@
@u
k
@f.X.U//
@x
j
:
(5.4.17)
Thesafestprocedureistowritetemporarily
g.X/D
@f.X/
@x
j
I
then(5.4.17)becomes
@g.X.U//
@u
k
D
Xn
sD1
@g.X.U//
@x
s
@x
s
.U/
@u
k
:
Since
@g
@x
s
D
@
2
f
@x
s
@x
j
;
thisyields
@
@u
k
@f
@x
k
D
Xn
sD1
@
2
f
@x
s
@x
j
@x
s
@u
k
:
Substitutingthisinto(5.4.16)yields
@
2
h
@u
k
@u
i
D
Xn
jD1
@f
@x
j
@
2
x
j
@u
k
@u
i
C
Xn
jD1
@x
j
@u
i
Xn
sD1
@
2
f
@x
s
@x
j
@x
s
@u
k
:
(5.4.18)
Tocomputeh
u
i
u
k
.U
0
/fromthisformula,weevaluatethepartialderivativesofx
1
,x
2
,
...,x
n
atU
0
andthoseoff atX
0
DX.U
0
/. Theformulaisvalidifx
1
,x
2
,...,x
n
and
theirfirstpartialderivativesaredifferentiableatU
0
andf,f
x
i
,f
x
2
,...,f
x
n
andtheirfirst
partialderivativesaredifferentiableatX
0
.
Insteadofmemorizing(5.4.18), youshouldunderstandhowitisderivedandusethe
method,ratherthantheformula,whencalculatingsecondpartialderivativesofcomposite
functions.Thesamemethodappliestothecalculationofhigherderivatives.
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346 Chapter5
Real-ValuedFunctionsofSeveralVariables
Example5.4.5
Supposethatf
x
andf
y
inExample5.4.2aredifferentiableonanopen
setSinR
2
.Differentiating(5.4.11)withrespecttoryields
@
2
h
@r2
Dcos
@
@r
@f
@x
Csin
@
@r
@f
@y
Dcos
@2f
@x2
@x
@r
C
@2f
@y@x
@y
@r
Csin
@2f
@x@y
@x
@r
C
@2f
@y2
@y
@r
(5.4.19)
if.x;y/2S.Since
@x
@r
Dcos;
@y
@r
Dsin; and
@
2
f
@x@y
D
@
2
f
@y@x
if.x;y/2S(Exercise5.3.21),(5.4.19)yields
@
2
h
@r2
Dcos
2
@
2
f
@x2
C2sincos
@
2
f
@x@y
Csin
2
@
2
f
@y2
:
Differentiating(5.4.11)withrespecttoyields
@2h
@@r
Dsin
@f
@x
Ccos
@f
@y
Ccos
@
@
@f
@x
Csin
@
@
@f
@y
Dsin
@f
@x
Ccos
@f
@y
Ccos
@
2
f
@x2
@x
@
C
@
2
f
@y@x
@y
@
Csin
@
2
f
@x@y
@x
@
C
@
2
f
@y2
@y
@
:
Since
@x
@
Drsin and
@y
@
Drcos;
itfollowsthat
@2h
@@r
Dsin
@f
@x
Ccos
@f
@y
rsincos
@2f
@x2
@2f
@y2
Cr.cos
2
sin
2
/
@
2
f
@x@y
:
TheMeanValueTheorem
Foracompositefunctionoftheform
h.t/Df.x
1
.t/;x
2
.t/;:::;x
n
.t//
wheretisarealvariable,x
1
,x
2
,...,x
n
aredifferentiableatt
0
,andf isdifferentiableat
X
0
DX.t
0
/,(5.4.8)takestheform
h
0
.t
0
/D
Xn
jD1
f
x
j
.X.t
0
//x
0
j
.t
0
/:
(5.4.20)
Thiswillbeusefulintheproofofthefollowingtheorem.
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Section5.4
TheChainRuleandTaylor’sTheorem
347
Theorem5.4.5(MeanValueTheoremforFunctions of
n
Variables)
Letf becontinuousatX
1
D.x
11
;x
21
;:::;x
n1
/andX
2
D.x
12
;x
22
;:::;x
n2
/anddif-
ferentiableonthelinesegmentLfromX
1
toX
2
:Then
f.X
2
/f.X
1
/D
Xn
iD1
f
x
i
.X
0
/.x
i2
x
i1
/D.d
X
0
f/.X
2
X
1
/
(5.4.21)
forsomeX
0
onLdistinctfromX
1
andX
2
.
Proof
AnequationofLis
XDX.t/DtX
2
C.1t/X
1
; 0t1:
Ourhypothesesimplythatthefunction
h.t/Df.X.t//
iscontinuousonŒ0;1anddifferentiableon.0;1/.Since
x
i
.t/Dtx
i2
C.1t/x
i1
;
(5.4.20)impliesthat
h
0
.t/D
Xn
iD1
f
x
i
.X.t//.x
i2
x
i1
/; 0<t<1:
Fromthemeanvaluetheoremforfunctionsofonevariable(Theorem2.3.11),
h.1/h.0/Dh
0
.t
0
/
forsomet
0
2.0;1/. Sinceh.1/Df.X
2
/andh.0/Df.X
1
/,thisimplies(5.4.21)with
X
0
DX.t
0
/.
Corollary5.4.6
Iff
x
1
;f
x
2
;...;f
x
n
areidenticallyzeroinanopenregionSofR
n
;
thenf isconstantinS:
Proof
WewillshowthatifX
0
andXareinS,thenf.X/Df.X
0
/.SinceSisanopen
region,Sispolygonallyconnected(Theorem5.1.20).Therefore,therearepoints
X
0
;X
1
;:::;X
n
DX
suchthatthelinesegmentL
i
fromX
i1
toX
i
isinS,1in.FromTheorem5.4.5,
f.X
i
/f.X
i1
/D
Xn
iD1
.d
e
X
i
f/.X
i
X
i1
/;
where
e
XisonL
i
andthereforeinS.Therefore,
f
x
i
.
e
X
i
/Df
x
2
.
e
X
i
/DDf
x
n
.
e
X
i
/D0;
348 Chapter5
Real-ValuedFunctionsofSeveralVariables
whichmeansthatd
e
X
i
f 0.Hence,
f.X
0
/Df.X
1
/DDf.X
n
/I
thatis,f.X/Df.X
0
/foreveryXinS.
HigherDifferentialsandTaylor’sTheorem
Supposethatfisdefinedinann-ballB
.X
0
/,with>0.IfX2B
.X
0
/,then
X.t/DX
0
Ct.XX
0
/2B
.X/; 0t1;
sothefunction
h.t/Df.X.t//
isdefinedfor0t1.FromTheorem5.4.3(seealso(5.4.20)),
h
0
.t/D
Xn
iD1
f
x
i
.X.t/.x
i
x
i0
/
iffisdifferentiableinB
.X
0
/,and
h
00
.t/D
Xn
jD1
@
@x
j
Xn
iD1
@f.X.t//
@x
i
.x
i
x
i0
/
!
.x
j
x
j0
/
D
n
X
i;jD1
@
2
f.X.t//
@x
j
@x
i
.x
i
x
i0
/.x
j
x
j0
/
iff
x
1
,f
x
2
,...,f
x
n
aredifferentiableinB
.X
0
/.Continuinginthisway,weseethat
h
.r/
.t/D
Xn
i
1
;i
2
;:::;i
r
D1
@
r
f.X.t//
@x
i
r
@x
i
r1
@x
i
1
.x
i
1
x
i
1
;0
/.x
i
2
x
i
2
;0
/.x
i
r
x
i
r
;0
/ (5.4.22)
ifallpartialderivativesoff oforderr1aredifferentiableinB
.X
0
/.
Thismotivatesthefollowingdefinition.
Definition5.4.7
Supposethatr1andallpartialderivativesoff oforderr1
aredifferentiableinaneighborhoodofX
0
. Thentherthdifferentialoff f atX
0
,denoted
byd
.r/
X
0
f,isdefinedby
d
.r/
X
0
f D
n
X
i
1
;i
2
;:::;i
r
D1
@
r
f.X
0
/
@x
i
r
@x
i
r1
@x
i
1
dx
i
1
dx
i
2
dx
i
r
;
(5.4.23)
wheredx
1
,dx
2
,...,dx
n
arethedifferentialsintroducedinSection5.3;thatis,dx
i
isthe
functionwhosevalueatapointinR
n
istheithcoordinateofthepoint.Forconvenience,
wedefine
.d
.0/
X
0
f/Df.X
0
/:
Noticethatd
.1/
X
0
f Dd
X
0
f.
Section5.4
TheChainRuleandTaylor’sTheorem
349
UndertheassumptionsofDefinition5.4.7,thevalueof
@
r
f.X
0
/
@x
i
r
@x
i
r1
@x
i
1
depends onlyonthenumberoftimesf isdifferentiatedwithrespecttoeachvariable,
andnotontheorderinwhichthedifferentiationsareperformed(Exercise5.3.22).Hence,
Exercise5.3.12impliesthat(5.4.23)canberewrittenas
d
.r/
X
0
f D
X
r
r
1
Šr
2
Šr
n
Š
@
r
f.X
0
/
@x
r
1
1
@x
r
2
2
@x
r
n
n
.dx
1
/
r
1
.dx
2
/
r
2
.dx
n
/
r
n
;
(5.4.24)
where
P
r
indicatessummationoverallorderedn-tuples.r
1
;r
2
;:::;r
n
/ofnonnegative
integerssuchthat
r
1
Cr
2
CCr
n
Dr
and@x
r
i
i
isomittedfromthe“denominators”ofalltermsin(5.4.24)forwhichr
i
D0. In
particular,ifnD2,
d
.r/
X
0
f D
Xr
jD0
r
j
!
@
r
f.x
0
;y
0
/
@x@yrj
.dx/
j
.dy/
rj
:
Example5.4.6
Let
f.x;y/D
1
1CaxCby
;
whereaandbareconstants.Then
@
r
f.x;y/
@x@yrj
D.1/
r
a
j
b
rj
.1CaxCby/rC1
;
so
d
.r/
X
0
f D
.1/
r
.1Cax
0
Cby
0
/rC1
r
X
jD0
r
j
!
a
j
b
rj
.dx/
j
.dy/
rj
D
.1/
r
.1Cax
0
Cby
0
/rC1
.adxCbdy/
r
if1Cax
0
Cby
0
¤0.
Example5.4.7
Let
f.X/Dexp
0
@
Xn
jD1
a
j
x
j
1
A
;
wherea
1
,a
2
,...,a
n
areconstants.Then
@
r
f.X/
@x
r
1
1
@x
r
2
2
@x
r
n
n
D.1/
r
a
r
1
1
a
r
2
2
a
r
n
n
exp
0
@
Xn
jD1
a
j
x
j
1
A
:
350 Chapter5
Real-ValuedFunctionsofSeveralVariables
Therefore,
.d
.r/
X
0
f/.ˆ/D.1/
r
X
r
r
1
Šr
2
Šr
n
Š
a
r
1
1
a
r
2
2
a
r
n
n
.dx
1
/
r
1
.dx
2
/r
2
.dx
n
/
r
n
!
exp
0
@
Xn
jD1
a
j
x
j0
1
A
D.1/
r
.a
1
dx
1
Ca
2
dx
2
CCa
n
dx
n
/
r
exp
0
@
Xn
jD1
a
j
x
j0
1
A
(Exercise5.3.12).
ThenexttheoremisanalogoustoTaylor’stheoremforfunctionsofonevariable(Theo-
rem2.5.4).
Theorem5.4.8(Taylor’sTheoremforFunctions of
n
Variables)
Suppose
thatf anditspartialderivativesoforderkaredifferentiableatX
0
andXinR
n
andon
thelinesegmentLconnectingthem:Then
f.X/D
Xk
rD0
1
.d
.r/
X
0
f/.XX/C
1
.kC1/Š
.d
.kC1/
e
X
f/.XX
0
/
(5.4.25)
forsome
e
XonLdistinctfromX
0
andX.
Proof
Define
h.t/Df.X
0
Ct.XX
0
//:
(5.4.26)
WithˆDXX
0
,ourassumptionsandthediscussionprecedingDefinition5.4.7imply
thath,h0,...,h.kC1/ existonŒ0;1.FromTaylor’stheoremforfunctionsofonevariable,
h.1/D
Xk
rD0
h
.r/
.0/
C
h
.kC1/
./
.kC1/Š
;
(5.4.27)
forsome2.0;1/.From(5.4.26),
h.0/Df.X
0
/ and h.1/Df.X/:
(5.4.28)
From(5.4.22)and(5.4.23)withˆDXX
0
,
h
.r/
.0/D.d
.r/
X
0
f/.XX
0
/; 1rk;
(5.4.29)
and
h
.kC1/
./D
d
kC1
e
X
f
.XX
0
/
(5.4.30)
Section5.4
TheChainRuleandTaylor’sTheorem
351
where
e
XDX
0
C.XX
0
/
isonLanddistinctfrom X
0
andX. Substituting(5.4.28), , (5.4.29), and(5.4.30)into
(5.4.27)yields(5.4.25).
Example5.4.8
Theorem5.4.8andtheresultsofExample5.4.6withX
0
D.0;0/and
ˆD.x;y/implythatif1CaxCby>0,then
1
1CaxCby
D
Xk
rD0
.1/
r
.axCby/
r
C.1/
kC1
.axCby/
kC1
.1CaxCby/kC2
forsome2.0;1/.(Notethatdependsonkaswellas.x;y/.)
Example5.4.9
Theorem 5.4.8andtheresultsofExample5.4.7withX
0
D 0and
ˆDXimplythat
exp
0
@
Xn
jD1
a
j
x
j
1
A
D
Xk
rD0
.1/
r
.a
1
x
1
Ca
2
x
2
CCa
n
x
n
/
r
C
.1/
kC1
.kC1/Š
.a
1
x
1
Ca
2
x
2
CCa
n
x
n
/
kC1
exp
2
4

0
@
Xn
jD1
a
j
x
j
1
A
3
5
;
forsome2.0;1/.
Byanalogywiththesituationforfunctionsofonevariable,wedefinethekthTaylor
polynomialoff aboutX
0
by
T
k
.X/D
k
X
rD0
1
.d
.r/
X
0
f/.XX
0
/
ifthedifferentialsexist;then(5.4.25)canberewrittenas
f.X/DT
k
.X/C
1
.kC1/Š
.d
.kC1/
e
X
f/.XX
0
/:
ASufficientConditionforRelativeExtremeValues
Thenexttheoremleadstoausefulsufficientconditionforlocalmaximaandminima. It
isrelatedtoTheorem2.5.1. Strictlyspeaking,however,itisnotageneralizationofTheo-
rem2.5.1(Exercise5.4.18).
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