﻿
322 Chapter5
Real-ValuedFunctionsofSeveralVariables
Takingthelimitash!0yields
jf
yx
.x
0
;y
0
/f
xy
.x
0
;y
0
/j:
Sinceisanarbitrarypositivenumber,thisproves(5.3.5).
Theorem5.3.3impliesthefollowingtheorem.Weleavetheprooftoyou(Exercises5.3.10
and5.3.11).
Theorem5.3.4
Supposethatfandallitspartialderivativesoforderrarecontin-
uousonanopensubsetSofR
n
:Then
f
x
i
1
x
i
2
;:::;x
i
r
.X/Df
x
j
1
x
j
2
;:::;x
j
r
.X/; X2S;
(5.3.11)
ifeachofthevariablesx
1
;x
2
;...;x
n
fx
i
1
;x
i
2
;:::;x
i
r
g and fx
j
1
;x
j
2
;:::;x
j
r
g:
Ifthisnumberisr
k
;wedenotethecommonvalueofthetwosidesof(5.3.11)by
@
r
f.X/
@x
r
1
1
@x
r
2
2
@x
r
n
n
;
(5.3.12)
itbeingunderstoodthat
0r
k
r; 1kn;
(5.3.13)
r
1
Cr
2
CCr
n
Dr;
(5.3.14)
and;ifr
k
D0;weomitthesymbol@x
0
k
fromthe“denominator”of(5.3.12):
Forexample,iff satisﬁesthehypothesesofTheorem5.3.4withkD4atapointX
0
in
R
n
(n2),then
f
xxyy
.X
0
/Df
xyxy
.X
0
/Df
xyyx
.X
0
/Df
yyxx
.X
0
/Df
yxyx
.X
0
/Df
yxxy
.X
0
/;
andtheircommonvalueisdenotedby
@4f.X
0
/
@x2@y2
:
Itcanbeshown(Exercise5.3.12)thatiff isafunctionof.x
1
;x
2
;:::;x
n
/and.r
1
;r
2
;:::;r
n
/
isaﬁxedorderedn-tuplethatsatisﬁes(5.3.13)and(5.3.14),thenthenumberofpartial
derivativesf
x
i
1
x
i
2
x
i
r
thatinvolvedifferentiationr
i
timeswithrespecttox
i
,1i n,
equalsthemultinomialcoefﬁcient
r
1
Šr
2
Šr
n
Š
:
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Section5.3
PartialDerivativesandtheDifferential
323
DiﬀerentiableFunctions ofSeveralVariables
Afunctionofseveralvariablesmayhaveﬁrst-orderpartialderivativesatapointX
0
butfail
tobecontinuousatX
0
.Forexample,if
f.x;y/D
(
xy
x2Cy2
; .x;y/¤.0;0/;
0;
.x;y/D.0;0/;
(5.3.15)
then
f
x
.0;0/Dlim
h!0
f.h;0/f.0;0/
h
D lim
h!0
00
h
D0
and
f
y
.0;0/Dlim
k!0
f.0;k/f.0;0/
k
D lim
k!0
00
k
D0;
butf isnotcontinousat.0;0/.(SeeExamples5.2.3and5.2.11.)Therefore,ifdifferentia-
bilityofafunctionofseveralvariablesistobeastrongerpropertythancontinuity,asitis
forfunctionsofonevariable,thedeﬁnitionofdifferentiabilitymustrequiremorethanthe
existenceofﬁrstpartialderivatives.Exercise2.3.1characterizesdifferentiabilityofafunc-
tionf ofonevariableinawaythatsuggeststhepropergeneralization:f isdifferentiable
atx
0
ifandonlyif
lim
x!x
0
f.x/f.x
0
/m.xx
0
/
xx
0
D0
forsomeconstantm,inwhichcasemDf
0
.x
0
/.
Thegeneralizationtofunctionsofnvariablesisasfollows.
Deﬁnition5.3.5
Afunctionf isdifferentiableat
X
0
D.x
10
;x
20
;:::;x
n0
//
ifX
0
2D
0
f
andthereareconstantsm
1
,m
2
,...;m
n
suchthat
lim
X!X
0
f.X/f.X
0
/
Xn
iD1
m
i
.x
i
x
i0
/
jXX
0
j
D0:
(5.3.16)
Example5.3.5
Let
f.x;y/Dx
2
C2xy:
Wewillshowthatf isdifferentiableatanypoint.x
0
;y
0
/,asfollows:
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324 Chapter5
Real-ValuedFunctionsofSeveralVariables
f.x;y/f.x
0
;y
0
/Dx
2
C2xyx
2
0
2x
0
y
0
Dx
2
x
2
0
C2.xyx
0
y
0
/
D.xx
0
/.xCx
0
/C2.xyx
0
y/C2.x
0
yx
0
y
0
/
D.xCx
0
C2y/.xx
0
/C2x
0
.yy
0
/
D2.x
0
Cy
0
/.xx
0
/C2x
0
.yy
0
/
C.xx
0
/.xx
0
C2y2y
0
/
Dm
1
.xx
0
/Cm
2
.yy
0
/C.xx
0
/.xx
0
C2y2y
0
/;
where
m
1
D2.x
0
Cy
0
/Df
x
.x
0
;y
0
/ and m
2
D2x
0
Df
y
.x
0
;y
0
/:
(5.3.17)
Therefore,
jf.x;y/f.x
0
;y
0
/m
1
.xx
0
/m
2
.yy
0
/j
jXX
0
j
D
jxx
0
jj.xx
0
/C2.yy
0
/j
jXX
0
j
p
5jXX
0
j;
bySchwarz’sinequality.Thisimpliesthat
lim
X!X
0
f.x;y/f.x
0
;y
0
/m
1
.xx
0
/m
2
.yy
0
/
jXX
0
j
D0;
sof isdifferentiableat.x
0
;y
0
/.
From(5.3.17),m
1
D f
x
.x
0
;y
0
/andm
2
D f
y
.x
0
;y
0
/inExample5.3.5. Thenext
theoremshowsthatthisisnotacoincidence.
Theorem5.3.6
Iff is s differentiableatX
0
D .x
10
;x
20
;:::;x
n0
/;thenf
x
1
.X
0
/;
f
x
2
.X
0
/;...;f
x
n
.X
0
/existandtheconstantsm
1
;m
2
;...;m
n
in(5.3.16)aregivenby
m
i
Df
x
i
.X
0
/; 1inI
(5.3.18)
thatis;
lim
X!X
0
f.X/f.X
0
/
Xn
iD1
f
x
i
.X
0
/.x
i
x
i0
/
jXX
0
j
D0:
Proof
Letibeagivenintegerinf1;2;:::;ng.LetXDX
0
CtE
i
,sothatx
i
Dx
i0
Ct,
x
j
Dx
j0
ifj ¤i,andjXX
0
jDjtj.Then(5.3.16)andthedifferentiabilityoff atX
0
implythat
lim
t!0
f.X
0
CtE
i
/f.X
0
/m
i
t
t
D0:
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Section5.3
PartialDerivativesandtheDifferential
325
Hence,
lim
t!0
f.X
0
CtE
i
/f.X
0
/
t
Dm
i
:
Thisproves(5.3.18),sincethelimitontheleftisf
x
i
.X
0
/,bydeﬁnition.
Alinearfunctionisafunctionoftheform
L.X/Dm
1
x
1
Cm
2
x
2
CCm
n
x
n
;
(5.3.19)
wherem
1
,m
2
,...;m
n
areconstants. FromDeﬁnition5.3.5,f f isdifferentiableatX
0
if
andonlyifthereisalinearfunctionLsuchthatf.X/f.X
0
/canbeapproximatedso
wellnearX
0
by
L.X/L.X
0
/DL.XX
0
/
that
f.X/f.X
0
/DL.XX
0
/CE.X/.jXX
0
j/;
(5.3.20)
where
lim
X!X
0
E.X/D0:
(5.3.21)
Theorem5.3.7
Iff isdifferentiableatX
0
;thenf iscontinuousatX
0
.
Proof
From(5.3.19)andSchwarz’sinequality,
jL.XX
0
/jMjXX
0
j;
where
MD.m
2
1
Cm
2
2
CCm
2
n
/
1=2
:
Thisand(5.3.20)implythat
jf.X/f.X
0
/j.MCjE.X/j/jXX
0
j;
which,with(5.3.21),impliesthatf iscontinuousatX
0
.
Theorem5.3.7impliesthatthefunctionf deﬁnedby(5.3.15)isnotdifferentiableat
.0;0/, sinceitisnotcontinuousat.0;0/. . However, , f
x
.0;0/andf
y
.0;0/exist, sothe
converseofTheorem5.3.7isfalse; thatis, , afunctionmayhavepartialderivativesata
pointwithoutbeingdifferentiableatthepoint.
TheDiﬀerential
Theorem 5.3.7impliesthatiff f isdifferentiableatX
0
, thenthereisexactlyonelinear
functionLthatsatisﬁes(5.3.20)and(5.3.21):
L.X/Df
x
1
.X
0
/x
1
Cf
x
2
.X
0
/x
2
CCf
x
n
.X
0
/x
n
:
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326 Chapter5
Real-ValuedFunctionsofSeveralVariables
Thisfunctioniscalledthedifferentialoff atX
0
. Wewilldenoteitbyd
X
0
f andits
valueby.d
X
0
f/.X/;thus,
.d
X
0
f/.X/Df
x
1
.X
0
/x
1
Cf
x
2
.X
0
/x
2
CCf
x
n
.X
0
/x
n
:
(5.3.22)
Intermsofthedifferential,(5.3.16)canberewrittenas
lim
X!X
0
f.X/f.X
0
/.d
X
0
f/.XX
0
/
jXX
0
j
D0:
Forconvenienceinwritingd
X
0
f,andtoconformwithstandardnotation,weintroduce
thefunctiondx
i
,deﬁnedby
dx
i
.X/Dx
i
I
thatis,dx
i
isthefunctionwhosevalueatapointinR
n
istheithcoordinateofthepoint.It
isthedifferentialofthefunctiong
i
.X/Dx
i
.From(5.3.22),
d
X
0
f Df
x
1
.X
0
/dx
1
Cf
x
2
.X
0
dx
2
CCf
x
n
.X
0
/dx
n
:
(5.3.23)
IfwewriteXD.x;y;:::;/,thenwewrite
d
X
0
f Df
x
.X
0
/dxCf
y
.X
0
/dyC;
wheredx,dy,... arethefunctionsdeﬁnedby
dx.X/Dx; dy.X/Dy;:::
WhenitisnotnecessarytoemphasizethespeciﬁcpointX
0
,(5.3.23)canbewrittenmore
simplyas
df Df
x
1
dx
1
Cf
x
2
dx
2
CCf
x
n
dx
n
:
Whendealingwithaspeciﬁcfunctionatanarbitrarypointofitsdomain,wemayusethe
hybridnotation
df Df
x
1
.X/dx
1
Cf
x
2
.X/dx
2
CCf
x
n
.X/dx
n
:
Example5.3.6
WesawinExample5.3.5thatthefunction
f.x;y/Dx
2
C2xy
isdifferentiableateveryXinR
n
,withdifferential
df D.2xC2y/dxC2xdy:
Toﬁndd
X
0
f withX
0
D.1;2/,wesetx
0
D1andy
0
D2;thus,
d
X
0
f D6dxC2dy
and
.d
X
0
f/.XX
0
/D6.x1/C2.y2/:
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Section5.3
PartialDerivativesandtheDifferential
327
Sincef.1;2/D5,thedifferentiabilityoffat.1;2/impliesthat
lim
.x;y/!.1;2/
f.x;y/56.x1/2.y2/
p
.x1/2C.y2/2
D0:
Example5.3.7
Thedifferentialofafunctionf Df.x/ofonevariableisgivenby
d
x
0
f Df
0
.x
0
/dx;
wheredxistheidentityfunction;thatis,
dx.t/Dt:
Forexample,if
f.x/D3x
2
C5x
3
;
then
df D.6xC15x
2
/dx:
Ifx
0
D1,then
d
x
0
f D9dx; .d
x
0
f/.xx
0
/D9.xC1/;
and,sincef.1/D2,
lim
x!1
f.x/C29.xC1/
xC1
D0:
Unfortunately,thenotationforthedifferentialis socomplicatedthatitobscures the
simplicityoftheconcept. The e peculiarsymbolsdf, dx,dy, etc., , were e introducedin
theearlystagesofthedevelopmentofcalculustorepresentverysmall(“inﬁnitesimal”)
incrementsinthevariables. However,inmodernusagetheyarenotquantitiesatall,but
linearfunctions.Thismeaningofthesymboldxdiffersfromitsmeaningin
R
b
a
f.x/dx,
whereitservesmerelytoidentifythevariableofintegration;indeed,someauthorsomitit
inthelattercontextandwritesimply
R
b
a
f.
Theorem5.3.7impliesthefollowinglemma,whichisanalogoustoLemma2.3.2. We
leavetheprooftoyou(Exercise5.3.13).
Lemma5.3.8
Iff isdifferentiableatX
0
;then
f.X/f.X
0
/D.d
X
0
f/.XX
0
/CE.X/jXX
0
j;
whereEisdeﬁnedinaneighborhoodofX
0
and
lim
X!X
0
E.X/DE.X
0
/D0:
Theorems5.3.2and5.3.7andthedeﬁnitionofthedifferentialimplythefollowing
theorem.
328 Chapter5
Real-ValuedFunctionsofSeveralVariables
Theorem5.3.9
Iff andgaredifferentiableatX
0
;thensoaref Cgandfg. . The
sameistrueoff=gifg.X
0
/¤0.Thedifferentialsaregivenby
d
X
0
.f Cg/Dd
X
0
fCd
X
0
g;
d
X
0
.fg/Df.X
0
/d
X
0
gCg.X
0
/d
X
0
f;
and
d
X
0
f
g
D
g.X
0
/d
X
0
f f.X
0
/d
X
0
g
Œg.X
0
/2
:
Thenexttheoremprovidesawidelyapplicablesufﬁcientconditionfordifferentiability.
Theorem5.3.10
Iff
x
1
;f
x
2
;...;f
x
n
existonaneighborhoodofX
0
andarecontin-
uousatX
0
;thenf isdifferentiableatX
0
:
Proof
LetX
0
D.x
10
;x
20
;:::;x
n0
/andsupposethat >0. Ourassumptionsimply
thatthereisaı>0suchthatf
x
1
;f
x
2
;:::;f
x
n
aredeﬁnedinthen-ball
S
ı
.X
0
/D
˚
X
ˇ
ˇ
jXX
0
j<ı
and
jf
x
j
.X/f
x
j
.X
0
/j< if jXX
0
j<ı; 1j j n:
(5.3.24)
LetXD.x
1
;x
;
:::;x
n
/beinS
ı
.X
0
/.Deﬁne
X
j
D.x
1
;:::;x
j
;x
jC1;0
;:::;x
n0
/; 1j j n1;
andX
n
DX.Thus,for1j n,X
j
differsfromX
j1
inthejthcomponentonly,and
thelinesegmentfromX
j1
toX
j
isinS
ı
.X
0
/.Nowwrite
f.X/f.X
0
/Df.X
n
/f.X
0
/D
Xn
jD1
Œf.X
j
/f.X
j1
/;
(5.3.25)
andconsidertheauxiliaryfunctions
g
1
.t/Df.t;x
20
;:::;x
n0
/;
g
j
.t/Df.x
1
;:::;x
j1
;t;x
jC1;0
;:::;x
n0
/; 2j j n1;
g
n
.t/Df.x
1
;:::;x
n1
;t/;
(5.3.26)
where,ineachcase,allvariablesexcepttaretemporarilyregardedasconstants.Since
f.X
j
/f.X
j1
/Dg
j
.x
j
/g
j
.x
j0
/;
themeanvaluetheoremimpliesthat
f.X
j
/f.X
j1
/Dg
0
j
.
j
/.x
j
x
j0
/;
Section5.3
PartialDerivativesandtheDifferential
329
where
j
isbetweenx
j
andx
j0
.From(5.3.26),
g
0
j
.
j
/Df
x
j
.
b
X
j
/;
where
b
X
j
isonthelinesegmentfromX
j1
toX
j
.Therefore,
f.X
j
/f.X
j1
/Df
x
j
.
b
X
j
/.x
j
x
j0
/;
and(5.3.25)impliesthat
f.X/f.X
0
/D
Xn
jD1
f
x
j
.
b
X
j
/.x
j
x
j0
/
D
n
X
jD1
f
x
j
.X
0
/.x
j
x
j0
/C
n
X
jD1
Œf
x
j
.
b
X
j
/f
x
j
.X
0
/.x
j
x
j0
/:
Fromthisand(5.3.24),
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
f.X/f.X
0
/
Xn
jD1
f
x
j
.X
0
/.x
j
x
j0
/
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ

Xn
jD1
jx
j
x
j0
jnjXX
0
j;
whichimpliesthatf isdifferentiableatX
0
.
Wesaythatf iscontinuouslydifferentiableonasubsetSofR
n
ifSiscontainedinan
opensetonwhichf
x
1
,f
x
2
,...;f
x
n
arecontinuous.Theorem5.3.10impliesthatsucha
functionisdifferentiableateachX
0
inS.
Example5.3.8
If
f.x;y/D
x
2
Cy
2
xy
;
then
f
x
.x;y/D
2x
xy
x
2
Cy
2
.xy/2
and f
y
.x;y/D
2y
xy
C
x
2
Cy
2
.xy/2
:
Sincef
x
andf
y
arecontinuouson
SD
˚
.x;y/
ˇ
ˇ
x¤y
;
f iscontinuouslydifferentiableonS.
Example5.3.9
TheconditionsofTheorem5.3.10arenotnecessaryfordifferentiabil-
ity;thatis,afunctionmaybedifferentiableatapointX
0
evenifitsﬁrstpartialderivatives
arenotcontinuousatX
0
.Forexample,let
f.x;y/D
8
<
:
.xy/
2
sin
1
xy
; x¤y;
0;
xDy: