Straightening theorem for vector fields

Last updated February 25, 2019

In differential calculus, the domain-straightening theorem states that, given a vector field $X$ on a manifold, there exist local coordinates $y_{1},\dots ,y_{n}$ such that $X=\partial /\partial y_{1}$ in a neighborhood of a point where $X$ is nonzero. The theorem is also known as straightening out of a vector field.

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 study of the area beneath a curve.

Vector field assignment of a vector to each point in a subset of Euclidean space

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

Manifold topological space that at each point resembles Euclidean space

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

The Frobenius theorem in differential geometry can be considered as a higher-dimensional generalization of this theorem.

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields in much the same way as an integral curve may be assigned to a single vector field. The theorem is foundational in differential topology and calculus on manifolds.

Proof

It is clear that we only have to find such coordinates at 0 in $\mathbb {R} ^{n}$ . First we write $X=\sum _{j}f_{j}(x){\partial \over \partial x_{j}}$ where $x$ is some coordinate system at $0$ . Let $f=(f_{1},\dots ,f_{n})$ . By linear change of coordinates, we can assume $f(0)=(1,0,\dots ,0).$ Let $\Phi (t,p)$ be the solution of the initial value problem ${\dot {x}}=f(x),x(0)=p$ and let

\psi (x_{1},\dots ,x_{n})=\Phi (x_{1},(0,x_{2},\dots ,x_{n})).

$\Phi$ (and thus $\psi$ ) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that

{\partial  \over \partial x_{1}}\psi (x)=f(\psi (x))

,

and, since $\psi (0,x_{2},\dots ,x_{n})=\Phi (0,(0,x_{2},\dots ,x_{n}))=(0,x_{2},\dots ,x_{n})$ , the differential $d\psi$ is the identity at $0$ . Thus, $y=\psi ^{-1}(x)$ is a coordinate system at $0$ . Finally, since $x=\psi (y)$ , we have: ${\partial x_{j} \over \partial y_{1}}=f_{j}(\psi (y))=f_{j}(x)$ and so ${\partial \over \partial y_{1}}=X$ as required.

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References

Theorem B.7 in Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke. Poisson Structures, Springer, 2013.

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