In differential calculus, the domain-straightening theorem states that, given a vector field
{\displaystyle X}
on a manifold, there exist local coordinates
y
/
y
in a neighborhood of a point where
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
First we write
is some coordinate system at
are the component function of
relative to
By linear change of coordinates, we can assume
be the solution of the initial value problem
ψ
) is smooth by smooth dependence on initial conditions in ordinary differential equations.
ψ ( 0 ,
, the differential
d ψ
is the identity at
ψ
is a coordinate system at
Finally, since
x = ψ ( y )
( ψ ( y ) ) =
as required.