pdfsharp c# : Chrome pdf from link software application cloud windows html wpf class TRENCH_REAL_ANALYSIS43-part263

422 Chapter6
Vector-ValuedFunctionsofSeveralVariables
HenceforthweassumethatX2N.Now,
.X;0/Dˆ..X;0//
(sinceˆD
1
/
Dˆ.X;G.X//
(since.X;0/D.X;G.X//)
D.X;F.X;G.X/// (sinceˆ.X;U/D.X;F.X;U//):
Therefore,F.X;G.X//D0;thatis,Gsatisfies(6.4.6). ToseethatGisunique,suppose
thatG
1
WR
n
!R
m
alsosatisfies(6.4.6).Then
ˆ.X;G.X//D.X;F.X;G.X///D.X;0/
and
ˆ.X;G
1
.X//D.X;F.X;G
1
.X///D.X;0/
forallXinN.Sinceˆisone-to-oneonM,thisimpliesthatG.X/DG
1
.X/.
Sincethepartialderivatives
@h
i
@x
j
; 1im; 1j j n;
arecontinuousfunctionsof.X;V/on
c
M, theyarecontinuouswithrespecttoXonthe
subset
˚
.X;0/
ˇ
ˇ
X2N
of
c
M.Therefore,GiscontinuouslydifferentiableonN.Toverify
(6.4.7),wewriteF.X;G.X//D0intermsofcomponents;thus,
f
i
.x
1
;x
2
;:::;x
n
;g
1
.X/;g
2
.X/;:::;g
m
.X//D0; 1i i m; X2N:
Sincef
i
andg
1
,g
2
,...,g
m
arecontinuouslydifferentiableontheirrespectivedomains,
thechainrule(Theorem5.4.3)impliesthat
@f
i
.X;G.X//
@x
j
C
Xm
rD1
@f
i
.X;G.X//
@u
r
@g
r
.X/
@x
j
D0; 1im; ; 1j j n; (6.4.11)
or,inmatrixform,
F
X
.X;G.X//CF
U
.X;G.X//G
0
.X/D0:
(6.4.12)
Since.X;G.X//2M forallXinN N andF
U
.X;U/isnonsingularwhen.X;U/2M,we
canmultiply(6.4.12)ontheleftbyF
1
U
.X;G.X//toobtain(6.4.7). Thiscompletesthe
proof.
InTheorem6.4.1wedenotedtheimplicitlydefinedtransformationbyG forreasons
ofclarityintheproof. However, , inapplyingthetheoremitisconvenienttodenotethe
transformationmoreinformallybyUDU.X/;thus,U.X
0
/DU
0
,andwereplace(6.4.6)
and(6.4.7)by
.X;U.X//2M
and X.X;U.X//D0 if X2N;
and
U
0
.X/DŒF
U
.X;U.X//
1
F
X
.X;U.X//; X2N;
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Section6.4
TheImplicitFunctionTheorem
423
while(6.4.11)becomes
@f
i
@x
j
C
m
X
rD1
@f
i
@u
r
@u
r
@x
j
D0; 1im;1j j n;
(6.4.13)
itbeingunderstoodthatthepartialderivativesofu
r
andf
i
areevaluatedatXand.X;U.X//,
respectively.
ThefollowingcorollaryistheimplicitfunctiontheoremformD1.
Corollary6.4.2
Supposethatf W W R
nC1
! Riscontinuouslydifferentiableonan
opensetcontaining.X
0
;u
0
/;withf.X
0
;u
0
/D 0andf
u
.X
0
;u
0
/¤0. Thenthereisa
neighborhoodMof.X
0
;u
0
/;containedinS;andaneighborhoodNofX
0
inR
n
onwhich
isdefinedauniquecontinuouslydifferentiablefunctionuDu.X/WR
n
!Rsuchthat
.X;u.X//2M
and f
u
.X;u.X//¤0; X2N;
u.X
0
/Du
0
; and f.X;u.X//D0; X2N:
Thepartialderivativesofuaregivenby
u
x
i
.X/D
f
x
i
.X;u.X//
f
u
.X;u.X//
; 1in:
Example6.4.1
Let
f.x;y;u/D1x
2
y
2
u
2
and.x
0
;y
0
;u
0
/D.
1
2
;
1
2
;
1
p
2
/.Thenf.x
0
;y
0
0
/D0and
f
x
.x;y;u/D2x; f
y
.x;y;u/D2y; f
u
.x;y;u/D2u:
Sincefiscontinuouslydifferentiableeverywhereandf
u
.x
0
;y
0
;u
0
/D
p
2¤0,Corol-
lary6.4.2impliesthattheconditions
1x
2
y
2
u
2
D0; u.1=2;1=2/D
1
p
2
;
determineuDu.x;y/near.x
0
;y
0
/D.
1
2
;
1
2
/sothat
u
x
.x;y/D
f
x
.x;y;u.x;y//
f
u
.x;y;u.x;y//
D
x
u.x;y/
;
(6.4.14)
and
u
y
.x;y/D
f
y
.x;y;u.x;y//
f
u
.x;y;u.x;y//
D
y
u.x;y/
:
(6.4.15)
Itisnotnecessarytomemorizeformulaslike(6.4.14)and(6.4.15).Sinceweknowthat
f anduaredifferentiable,wecanobtain(6.4.14)and(6.4.15)byapplyingthechainrule
totheidentity
f.x;y;u.x;y//D0:
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424 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Example6.4.2
Let
f.x;y;u/Dx
3
y
2
u
2
C3xy
4
u
4
3x
6
y
6
u
7
C12x13
(6.4.16)
and.x
0
;y
0
;u
0
/D.1;1;1/,sof.x
0
;y
0
;u
0
/D0.Then
f
x
.x;y;u/D3x
2
y
2
u
2
C3y
4
u
4
18x
5
y
6
u
7
C12;
f
y
.x;y;u/D2x
3
yu
2
C12xy
3
u
4
18x
6
y
5
u
7
;
f
u
.x;y;u/D2x
3
y
2
uC12xy
4
u
3
21x
6
y
6
u
6
:
Sincef
u
.1;1;1/D7¤0,Corollary6.4.2impliesthattheconditions
f.x;y;u/D0; u.1;1/D1
(6.4.17)
determineuasacontinuouslydifferentiablefunctionof.x;y/near.1;1/.
Ifwetrytosolve(6.4.16)foru, weseeveryclearlythatTheorem6.4.1andCorol-
lary6.4.2areexistencetheorems;thatis,theytellusthatthereisafunctionuDu.x;y/
thatsatisfies(6.4.17),butnothowtofindit.Inthiscasethereisnoconvenientformulafor
thefunction,althoughitspartialderivativescanbeexpressedconvenientlyintermsofx,
y,andu.x;y/:
u
x
.x;y/D
f
x
.x;y;u.x;y//
f
u
.x;y;u.x;y//
; u
y
.x;y/D
f
y
.x;y;u.x;y//
f
u
.x;y;u.x;y//
:
Inparticular,sinceu.1;1/D1,
u
x
.1;1/D
0
7
D0; u
y
.1;1/D
4
7
D
4
7
:
Example6.4.3
Let
XD
2
4
x
y
´
3
5
and UD
u
v
;
and
F.X;U/D
2x
2
Cy
2
2
Cu
2
v
2
x
2
2
C2uv
:
IfX
0
D.1;1;1/andU
0
D.0;2/,thenF.X
0
;U
0
/D0.Moreover,
F
U
.X;U/D
2u 2v
2
1
and F
X
D
4x 2y 2´
2x
0
;
so
det.F
U
.X
0
;U
0
//D
ˇ
ˇ
ˇ
ˇ
0 4
2 1
ˇ
ˇ
ˇ
ˇ
D8¤0:
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Section6.4
TheImplicitFunctionTheorem
425
Hence,theconditions
F.X;U/D0; U.1;1;1/D.0;2/
determineU D U.X/nearX
0
. AlthoughitisdifficulttofindU.X/explicitly,wecan
approximateU.X/nearX
0
byanaffinetransformation.Thus,from(6.4.7),
U
0
.X
0
/DŒF
U
.X
0
;U.X
0
//
1
F
X
.X
0
;U.X
0
//
(6.4.18)
D
0 4
2 1
1
4 2 2
2
0 2
D
1
8
1 4
2 0

4 2 2
2
0 2
D
1
8
4 2
6
8 4 4
:
Therefore,
lim
X!.1;1;1/
u.x;y/
v.x;y/
0
2
C
1
8
4 2
6
8 4 4
2
4
x1
yC1
´1
3
5
Œ.x1/2C.yC1/2C.´1/2
1=2
D
0
0
:
Again,itisnotnecessarytomemorize(6.4.18),sincethepartialderivativesofanimplic-
itlydefinedfunctioncanbeobtainedfromthechainruleandCramer’srule,asinthenext
example.
Example6.4.4
LetuDu.x;y/andvDv.x;y/bedifferentiableandsatisfy
x2C2yC3´2CuCvD6
2x
3
C4y
2
C2´
2
CuCv
2
D9
(6.4.19)
and
u.1;1;0/D1; v.1;1;0/D2:
(6.4.20)
Tofindu
x
andv
x
,wedifferentiate(6.4.19)withrespecttoxtoobtain
2xC2uu
x
Cv
x
D0
6x
2
Cu
x
C2vv
x
D0:
Therefore,
2u
1
1 2v

u
x
v
x
D
2x
6x
2
;
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426 Chapter6
Vector-ValuedFunctionsofSeveralVariables
andCramer’sruleyields
u
x
D
ˇ
ˇ
ˇ
ˇ
2x
1
6x
2
2v
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2u
1
1 2v
ˇ
ˇ
ˇ
ˇ
D
6x
2
4xv
4uv1
and
v
x
D
ˇ
ˇ
ˇ
ˇ
2u
2x
1 6x
2
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2u
1
1 2v
ˇ
ˇ
ˇ
ˇ
D
2x12x
2
u
4uv1
if4uv¤1.Inparticular,from(6.4.20),
u
x
.1;1;0/D
2
9
D
2
9
; v
x
.1;1;0/D
14
9
D
14
9
:
Jacobians
ItisconvenienttoextendthenotationintroducedinSection6.2fortheJacobianofatrans-
formationF W R! Rm. Iff
1
,f
2
,..., f
m
arereal-valuedfunctionsofkvariables,
km,and
1
,
2
,...,
m
areanymofthevariables,thenwecallthedeterminant
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
@f
1
@
1
@f
1
@
2

@f
1
@
m
@f
2
@
1
@f
2
@
2

@f
2
@
m
:
:
:
:
:
:
:
:
:
:
:
:
@f
m
@
1
@f
m
@
2

@f
m
@
m
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
;
theJacobianoff
1
,f
2
,...,f
m
withrespectto
1
,
2
,...,
m
.WedenotethisJacobianby
@.f
1
;f
2
;:::;f
m
/
@.
1
;
2
;:::;
m
/
;
andwedenotethevalueoftheJacobianatapointPby
@.f
1
;f
2
;:::;f
m
/
@.
1
;
2
;:::;
m
/
ˇ
ˇ
ˇ
ˇ
ˇ
P
:
Example6.4.5
If
F.x;y;´/D
3x
2
C2xyC´
2
4x
2
C2xy
2
3
;
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Section6.4
TheImplicitFunctionTheorem
427
then
@.f
1
;f
2
/
@.x;y/
D
ˇ
ˇ
ˇ
ˇ
6xC2y
2x
8xC2y
2
4xy
ˇ
ˇ
ˇ
ˇ
;
@.f
1
;f
2
/
@.y;´/
D
ˇ
ˇ
ˇ
ˇ
2x
4xy 3´
2
ˇ
ˇ
ˇ
ˇ
;
and
@.f
1
;f
2
/
@.´;x/
D
ˇ
ˇ
ˇ
ˇ
6xC2y
2 8xC2y2
ˇ
ˇ
ˇ
ˇ
:
ThevaluesoftheseJacobiansatX
0
D.1;1;0/are
@.f
1
;f
2
/
@.x;y/
ˇ
ˇ
ˇ
ˇ
ˇ
X
0
D
ˇ
ˇ
ˇ
ˇ
4 2
6 4
ˇ
ˇ
ˇ
ˇ
D4;
@.f
1
;f
2
/
@.y;´/
ˇ
ˇ
ˇ
ˇ
ˇ
X
0
D
ˇ
ˇ
ˇ
ˇ
2 0
4 0
ˇ
ˇ
ˇ
ˇ
D0;
and
@.f
1
;f
2
/
@.´;x/
ˇ
ˇ
ˇ
ˇ
ˇ
X
0
D
ˇ
ˇ
ˇ
ˇ
0 4
0 6
ˇ
ˇ
ˇ
ˇ
D0:
TherequirementinTheorem6.4.1thatF
U
.X
0
;U
0
/benonsingularisequivalentto
@.f
1
;f
2
;:::;f
m
/
@.u
1
;u
2
;:::;u
m
/
ˇ
ˇ
ˇ
ˇ
ˇ
.X
0
;U
0
/
¤0:
Ifthisissothen,forafixedj,Cramer’sruleallowsustowritethesolutionof(6.4.13)as
@u
i
@x
j
D
@.f
1
;f
2
;:::;f
i
;:::;f
m
/
@.u
1
;u
2
;:::;x
j
;:::;u
m
/
@.f
1
;f
2
;:::;f
i
;:::;f
m
/
@.u
1
;u
2
;:::;u
i
;:::;u
m
/
; 1im;
Noticethatthedeterminantinthenumeratorontherightisobtainedbyreplacingtheith
columnofthedeterminantinthedenominator,whichis
2
6
6
6
6
6
6
6
6
6
4
@f
1
@u
i
@f
2
@u
i
:
:
:
@f
m
@u
i
3
7
7
7
7
7
7
7
7
7
5
; by
2
6
6
6
6
6
6
6
6
6
4
@f
1
@x
j
@f
2
@x
j
:
:
:
@f
m
@x
j
3
7
7
7
7
7
7
7
7
7
5
:
Sofarwe haveconsideredonlytheproblemofsolvingacontinuouslydifferentiable
system
F.X;U/D0 .FWR
nCm
!R
m
/
(6.4.21)
forthelastmvariables,u
1
,u
2
,...,u
m
,intermsofthefirstn,x
1
,x
2
,...,x
n
. Thiswas
merelyforconvenience;(6.4.21)canbesolvednear.X
0
;U
0
/foranymofthevariablesin
termsoftheothern,providedonlythattheJacobianoff
1
,f
2
,...,f
m
withrespecttothe
chosenmvariablesisnonzeroat.X
0
;U
0
/.Thiscanbeseenbyrenamingthevariablesand
applyingTheorem6.4.1.
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428 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Example6.4.6
Let
F.x;y;´/D
f.x;y;´/
g.x;y;´/
becontinuouslydifferentiableinaneighborhoodof.x
0
;y
0
0
/.Supposethat
F.x
0
;y
0
0
/D0
and
@.f;g/
@.x;´/
ˇ
ˇ
ˇ
ˇ
ˇ
.x
0
;y
0
0
/
¤0:
(6.4.22)
ThenTheorem6.4.1withXD.y/andUD.x;´/impliesthattheconditions
f.x;y;´/D0; g.x;y;´/D0; x.y
0
/Dx
0
; ´.y
0
/D´
0
;
(6.4.23)
determinexand´ascontinuouslydifferentiablefunctionsofyneary
0
. Differentiating
(6.4.23)withrespecttoyandregardingxand´asfunctionsofyyields
f
x
x
0
Cf
y
Cf
´
´
0
D0
g
x
x
0
Cg
y
Cg
´
´
0
D0:
Rewritingthisas
f
x
x
0
Cf
´
´
0
Df
y
g
x
x
0
Cg
´
´
0
Dg
y
;
andsolvingforx
0
and´
0
byCramer’sruleyields
x
0
D
ˇ
ˇ
ˇ
ˇ
f
y
f
´
g
y
g
´
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
f
x
f
´
g
x
g
´
ˇ
ˇ
ˇ
ˇ
D
@.f;g/
@.y;´/
@.f;g/
@.x;´/
(6.4.24)
and
´
0
D
ˇ
ˇ
ˇ
ˇ
f
x
f
y
g
x
g
y
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
f
x
f
´
g
x
g
´
ˇ
ˇ
ˇ
ˇ
D
@.f;g/
@.x;y/
@.f;g/
@.x;´/
:
(6.4.25)
Equation(6.4.22)impliesthat@.f;g/=@.x;´/isnonzeroifyissufficientlyclosetoy
0
.
Example6.4.7
LetX
0
D.1;1;2/and
F.x;y;´/D
f.x;y;´/
g.x;y;´/
D
6xC6yC4´344
x2y2C8´14
:
Section6.4
TheImplicitFunctionTheorem
429
ThenF.X
0
/D0,
@.f;g/
@.x;´/
D
ˇ
ˇ
ˇ
ˇ
6 12´
2
2x
8
ˇ
ˇ
ˇ
ˇ
;
and
@.f;g/
@.x;´/
ˇ
ˇ
ˇ
ˇ
ˇ
.1;1;2/
D
ˇ
ˇ
ˇ
ˇ
6 48
2
8
ˇ
ˇ
ˇ
ˇ
D144¤0:
Therefore,Theorem6.4.1withXD.y/andUD.x;´/impliesthattheconditions
f.x;y;´/D0; g.x;y;´/D0;
and
x.1/D1; ´.1/D2;
(6.4.26)
determinexand´ascontinuouslydifferentiablefunctionsofyneary
0
D1.From(6.4.24)
and(6.4.25),
x
0
D
@.f;g/
@.y;´/
@.f;g/
@.x;´/
D
ˇ
ˇ
ˇ
ˇ
6
12´
2
2y
8
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
6
12´2
2x
8
ˇ
ˇ
ˇ
ˇ
D
2Cy´
2
2Cx´2
and
´
0
D
@.f;g/
@.x;y/
@.f;g/
@.x;´/
D
ˇ
ˇ
ˇ
ˇ
6
6
2x 2y
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
6
12´
2
2x
8
ˇ
ˇ
ˇ
ˇ
D
yx
4C2x´2
:
TheseequationsholdnearyD1.Togetherwith(6.4.26)theyimplythat
x
0
.1/D1; ´
0
.1/D0:
Example6.4.8
ContinuingwithExample6.4.7,Theorem6.4.1impliesthatthecon-
ditions
f.x;y;´/D0; g.x;y;´/D0; y.1/D1; ´.1/D2
determineyand´asfunctionsofxnearx
0
D1,since
@.f;g/
@.y;´/
D
ˇ
ˇ
ˇ
ˇ
6
12´
2
2y
8
ˇ
ˇ
ˇ
ˇ
and
@.f;g/
@.y;´/
ˇ
ˇ
ˇ
ˇ
ˇ
.1;1;2/
D
ˇ
ˇ
ˇ
ˇ
6 48
2
8
ˇ
ˇ
ˇ
ˇ
D144¤0:
However,Theorem6.4.1doesnotimplythattheconditions
f.x;y;´/D0; g.x;y;´/D0; x.2/D1; y.2/D1
430 Chapter6
Vector-ValuedFunctionsofSeveralVariables
definexandyasfunctionsof´near´
0
D2,since
@.f;g/
@.x;y/
D
ˇ
ˇ
ˇ
ˇ
6
6
2x 2y
ˇ
ˇ
ˇ
ˇ
and
@.f;g/
@.x;y/
ˇ
ˇ
ˇ
ˇ
ˇ
.1;1;2/
D
ˇ
ˇ
ˇ
ˇ
6
6
2 2
ˇ
ˇ
ˇ
ˇ
D0:
Weclosethissectionbyobservingthatthefunctionsu
1
,u
2
,...,u
m
definedinTheo-
rem6.4.1havehigherderivativesiff
1
;f
2
;:::;f
m
do,andtheymaybeobtainedbydiffer-
entiating(6.4.13),usingthechainrule.(Exercise6.4.17).
Example6.4.9
Supposethatuandvarefunctionsof.x;y/thatsatisfy
f.x;y;u;v/Dxu
2
v
2
C9D0
g.x;y;u;v/Dyu
2
Cv
2
10D0:
Then
@.f;g/
@.u;v/
D
ˇ
ˇ
ˇ
ˇ
2u 2v
2u
2v
ˇ
ˇ
ˇ
ˇ
D8uv:
FromTheorem6.4.1,ifuv¤0,then
u
x
D
1
8uv
@.f;g/
@.x;v/
D
1
8uv
ˇ
ˇ
ˇ
ˇ
1 2v
0
2v
ˇ
ˇ
ˇ
ˇ
D
1
4u
;
u
y
D
1
8uv
@.f;g/
@.y;v/
D
1
8uv
ˇ
ˇ
ˇ
ˇ
0 2v
1
2v
ˇ
ˇ
ˇ
ˇ
D
1
4u
;
v
x
D
1
8uv
@.f;g/
@.u;x/
D
1
8uv
ˇ
ˇ
ˇ
ˇ
2u 1
2u 0
ˇ
ˇ
ˇ
ˇ
D
1
4v
;
v
y
D
1
8uv
@.f;g/
@.u;y/
D
1
8uv
ˇ
ˇ
ˇ
ˇ
2u 0
2u 1
ˇ
ˇ
ˇ
ˇ
D
1
4v
:
Thesecanbedifferentiatedasmanytimesaswewish.Forexample,
u
xx
D
u
x
4u2
D
1
16u3
;
u
xy
D
u
y
4u2
D
1
16u3
;
and
v
yx
D
v
x
4v2
D
1
16v2
:
Section6.4
TheImplicitFunctionTheorem
431
6.4Exercises
1.
SolveforUD.u;:::/asafunctionofXD.x;:::/.
(a)
1
1
1 1

u
v
C
1 1
2 3

x
y
D
0
0
(b)
uvCwC3xC2yD0
uCvCw xC yD0
uCvw
C yD0
(c)
3uC vCyDsinx
uC2vCxDsiny
(d)
2uC2vC wC2xC2yC ´D0
u vC2wC x yC2´D0
3uC2v wC3xC2y ´D0
2.
SupposethatX
0
2RandU
0
2Rm. Prove:IfN
1
isaneighborhoodof.X
0
;U
0
/
inRnCm,thereisaneighborhoodNofX
0
inRnsuchthat.X;U
0
/2N
1
ifX2N.
3.
Let.X
0
;U
0
/beanarbitrarypointinR
nCm
. GiveanexampleofafunctionF F W
R
nCm
!R
m
suchthatFiscontinuouslydifferentiableonR
nCm
,F.X
0
;U
0
/D0,
F
U
.X
0
;U
0
/issingular,andtheconditionsF.X;U/D0andU.X
0
/DY
0
(a)
determineUasacontinuouslydifferentiablefunctionofXforallX;
(b)
determineUasacontinuousfunctionofXforallX,butUisnotdifferentiable
atX
0
;
(c)
donotdetermineUasafunctionofX.
4.
LetuDu.x;y/bedeterminednear.1;1/by
x
2
yuC2xy
2
u
3
3x
3
y
3
u
5
D0; u.1;1/D1:
Findu
x
.1;1/andu
y
.1;1/.
5.
LetuDu.x;y;´/bedeterminednear.1;1;1/by
x
2
y
5
´
2
u
5
C2xy
2
u
3
3x
3
´
2
uD0; u.1;1;1/D1:
Findu
x
.1;1;1/,u
y
.1;1;1/,andu
´
.1;1;1/.
6.
Findu.x
0
;y
0
/,u
x
.x
0
;y
0
/,andu
y
.x
0
;y
0
/.
(a)
2x
2
Cy
2
Cue
u
D6; .x
0
;y
0
/D.1;2/
(b)
u.xC1/Cx.yC2/Cy.u2/D0; .x
0
;y
0
/D.1;2/
(c)
1e
u
sin.xCy/D0; .x
0
;y
0
/D.=4;=4/
(d)
xloguCylogxCulogyD0; .x
0
;y
0
/D.1;1/
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