differential calculus In mathematics, differential calculus is a subfield of calculus that studies 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. ...

, the domain-straightening theorem states that, given a

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

X

on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

, 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. The Frobenius theorem in

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

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

\mathbb^n

. First we write

X = \sum_j f_j(x)

where

x

is some coordinate system at

0,

and

f_1, f_2, \dots, f_n

are the component function of

X

relative to

x.

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 In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...

. It follows that :

\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^(x)

is a coordinate system at

0

. Finally, since

x = \psi(y)

, we have:

= f_j(\psi(y)) = f_j(x)

and so

= X

as required.

References

*Theorem B.7 in Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke. ''Poisson Structures'', Springer, 2013. {{Refend Differential calculus Theorems in mathematical analysis