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8
TadeuszKaczorek
∂
g
=
⎡
⎢
⎢
⎢
⋅
∂
∂
⋅
g
x
⋅
⋅
1
1
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
∂
∂
⋅
g
x
⋅
⋅
n
1
⋅
⋅
⎤
⎥
⎥
⎥
,
∂
f
=
⎡
⎢
⎢
⎢
⋅
∂
∂
⋅
x
f
⋅
1
1
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
∂
⋅
∂
⋅
x
f
⋅
1
n
⋅
⋅
⎤
⎥
⎥
⎥
,
∂
x
⎢
⎢
⎢
⎣
∂
∂
g
x
1
n
⋅
⋅
⋅
∂
∂
g
x
n
n
⎥
⎥
⎥
⎦
∂
x
⎢
⎢
⎢
⎣
∂
∂
f
x
n
1
⋅
⋅
⋅
∂
∂
x
f
n
n
⎥
⎥
⎥
⎦
Asubspace
Δ
()
x⊂
R
n
spannedonthevectors
f
1
()
x
,...,
f
m
()
x
,i.e.
Δ
()
x
=
span
{
f
1
()
x
,...,
f
m
()
x
}
(2)
iscalledthedistributioninthepoint
x
∈
U
⊂
R
n
.
Thedistribution
Δ
()
x
iscalledinvolutiveiftheLiebracket
[
f,
g
]
ofeverypair
ofvectorfieldsfandgbelongingto
Δ
()
x
belongsalsoto
Δ
()
x
.
Thedistribution(2)isinvolutiveifandonlyif
[
f
i
,
f
j
]
∈
Δ
()
x
(3)
forall
1
≤,
i
j
≤
m
.
Thecondition(3)issatisfiedifandonlyif[1,2,4]
rank
{
f
1
()
x
,...,
f
m
()
x
}
=
rank
{
f
1
()
x
,...,
f
m
()
x
,
[
f
i
,
f
j
]
}
(4)
forall
1
≤,
i
j
≤
m
.
Let
V
()
x
=
V
(
x
1
...,
x
n
)
bethepotentialofthefielddependingonthevector
x
=
[
x
1
,...,
x
n
]
T
,whosecomponents
x
i
=
x
i
()
t
,=
i
1
,...,
n
arefunctionsoftimet,
andTdenotesthetranspose.
Let
m
=n
−
1
vectorfields
f
i
()
x
∈
R
n
,=
i
1
,...,
m
,begiven,whichdefinea
nonsingulardistributionatthepoint
x
∈
U
⊂
R
n
oftheform(2).
Wearelookingforthepotential
V=
V
()
x
,whosethegradient
∂
∂
V
x
=
⎡
⎢
⎣
∂
∂
V
x
1
,
∂
∂
x
V
2
,
⋅
⋅
⋅
,
∂
∂
x
V
n
⎤
⎥
⎦
(5)
isortogonaltoallmvectorfields,i.e.
∂
∂
V
x
[
f
1
()
x
,...,
f
m
()
x
]
=
0
(6)
Theproblemundertheconsiderationscanbestatedasfollows.
Givenmvectorfields
f
i
()
x
∈
R
n
,=
i
1
,...,
m
,findapotentialV(x)ofthefield
satisfyingthepartialdifferentialequation(6)andthegiveninitialconditions
x
i
()
0
=
x
i
0
(7)
for
i
=
1
,...,
n
.
