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POZNANUNIVERSITYOFTECHNOLOGYACADEMICJOURNALS
No56
ElectricalEngineering
2007
TadeuszKACZOREK*
APPLICATIONOFTHEFROBENIUSTHEOREM
TODETERMINATIONOFPOTENTIALSWITH
THEIRGRADIENTSORTHOGONALTOVECTORFIELDS
Theproblemoffindingpotentialsofthevectorfieldthosegradientsareorthogonalto
thegivenvectorfieldsisformulatedandsolved.Thesolutiontotheproblemisbasedonthe
Frobeniustheorem.Itisshownthattheproblemhasasolutionifandonlyifthedistribution
spannedonthevectorfieldsisinvolutive.Aprocedureforfindingofthedesiredpotentials
isproposedanddemonstratedonanexampleoffindingonepotentialfortwogivenvector
fields.
1.INTRODUCTION
Inrecentyearsthemethodsofdifferentialgeometryhavebeenappliedto
analysisandsynthesisofnon-linearsystems[1,2,6].Usingthesemethodssome
knownresultsforlinearsystemshavebeenextendedfornon-linearsystems.
Theproblemofgloballinearizationofnon-linearsystemshasbeeninvestigated
inpapers[3-5].
InthisnotetheFrobeniustheoremwillbeappliedtosolvetheproblemof
findingpotentialsofthevectorfieldwhosegradientsareortogonaltothegiven
vectorfields.Necessaryandsufficientconditionsfortheexistanceofsolutionto
theproblemwillbeestablishedandaprocedureforfindingthesolutiontothe
problemwillbeproposed.
2.PRELIMINARIESANDPROBLEMFORMULATION
Considertwosmooth(ie.havingcontinuousderivativesofarbitraryorder)n-
dimensionalvectorfields
f=
f
()
x
and
g=
g
()
x
ofthevector
x∈
R
n
.Avector
fielddefinedbytheformula
[
f
,
g
]
=
∂
∂
g
x
f
−
∂
∂
f
x
g
(1)
iscalledtheLiebracketofthevectorfieldsfig,where
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*BialystokTechnicalUniversity.
