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ApplicationoftheFrobeniustheoremtodeterminationofpotentials...
9
InthisnoteusingtheFrobeniustheoremnecessaryandsufficientconditionsfor
theexistenceofsolutiontotheproblemwillbeestablishedandaprocedurefor
findingasolutiontotheproblemwillbeproposed.
3.PROBLEMSOLUTION
Necessaryandsufficientconditionsfortheexistenceofsolutiontotheproblem
followfromtheFrobeniustheorem[1,2].
Theorem1:Theequation(6)hasa(smooth)solutionV(x)ifandonlyifthe
distribution(2)isinvolutive.
FromtheconstructiveproofoftheTheorem1wehavethefollowingprocedure
forfindingthedesiredsolutionV(x)ofthepartialdifferentialequation(6).
Procedure
Step1
Usingthecondition(4)checkwhetherthedistribution(2)isinvolutive.Ifthe
distribution(2)isnotinvolutive,thentheequation(6)hasnotasolution.
Step2
Forgiven
f
i
()
x
∈
R
n
,
i
=
1
,...,
m
choose
f
n
()
x
suchthat
span
{
f
1
()
x
,...,
f
m
()()
x
,
f
n
x
}
=
R
n
(8)
Step3
Findsolution
φ
t
f
k
()
x
oftheequations
x
&
=
f
k
()
x
,
k
=
1
,...,
n
(9)
withinitialconditions(7),
x
()
0
=
x
0
.
Step4
Knowingthesolutions
φ
t
f
k
()
x
k
=
1
,...,
n
findthemap
w
(
z
1
,...,
z
n
)
=
φ
z
f
1
1
o
...
o
φ
z
f
n
n
()
x
0
(10)
whereodenotesthecompositionofthemaps.
Step5
Findtheinversemap
w
−
1
(
z
1
,...,
z
n
)
=
φ
()
x
=
⎡
⎢
⎢
⎢
⎣
φ
φ
1
n
...
()
()
x
x
⎤
⎥
⎥
⎥
⎦
(11)
andthedesiredsolution
V
()
x
=
φ
n
()
x
.
Theprocedurewillbedemonstratedofthefollowingexample.
