Domain-straightening theorem
Encyclopedia
In differential calculus
, the domain-straightening theorem states that, given a vector field
on a manifold
, there exist local coordinates
such that 
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.
. First we write 
where 
is some coordinate system at 
. Let 
. By linear change of coordinates, we can assume 
Let 
be the solution of the initial value problem 
and let
(and thus 
) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that
,
and, since
, the differential 
is the identity at 
. Thus, 
is a coordinate system at 
. Finally, since 
, we have: 
and so 
as required.
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
, the domain-straightening theorem states that, given a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
on a manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, there exist local coordinates
such that in a neighborhood of a point where is nonzero. The theorem is also known as straightening out of a vector field.The Frobenius theorem
Frobenius theorem (differential topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
in differential geometry can be considered as a higher dimensional generalization of this theorem.
Proof
It is clear that we only have to find such coordinates at 0 in. First we write where is some coordinate system at . Let . By linear change of coordinates, we can assume Let be the solution of the initial value problem and let (and thus ) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that,and, since
, the differential is the identity at . Thus, is a coordinate system at . Finally, since , we have: and so as required.