In differential calculus, the domain-straightening theorem states that, given a vector field
X
on a manifold, there exist local coordinates
y
1
,
…
,
y
n
such that
X
=
∂
/
∂
y
1
in a neighborhood of a point where
X
is nonzero. The theorem is also known as straightening out of a vector field.
The Frobenius theorem in differential geometry can be considered as a higher-dimensional generalization of this theorem.
It is clear that we only have to find such coordinates at 0 in
R
n
. First we write
X
=
∑
j
f
j
(
x
)
∂
∂
x
j
where
x
is some coordinate system at
0
. Let
f
=
(
f
1
,
…
,
f
n
)
. By linear change of coordinates, we can assume
f
(
0
)
=
(
1
,
0
,
…
,
0
)
.
Let
Φ
(
t
,
p
)
be the solution of the initial value problem
x
˙
=
f
(
x
)
,
x
(
0
)
=
p
and let
ψ
(
x
1
,
…
,
x
n
)
=
Φ
(
x
1
,
(
0
,
x
2
,
…
,
x
n
)
)
.
Φ
(and thus
ψ
) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that
∂
∂
x
1
ψ
(
x
)
=
f
(
ψ
(
x
)
)
,
and, since
ψ
(
0
,
x
2
,
…
,
x
n
)
=
Φ
(
0
,
(
0
,
x
2
,
…
,
x
n
)
)
=
(
0
,
x
2
,
…
,
x
n
)
, the differential
d
ψ
is the identity at
0
. Thus,
y
=
ψ
−
1
(
x
)
is a coordinate system at
0
. Finally, since
x
=
ψ
(
y
)
, we have:
∂
x
j
∂
y
1
=
f
j
(
ψ
(
y
)
)
=
f
j
(
x
)
and so
∂
∂
y
1
=
X
as required.
