A diffeomorphism is a smooth mapping whose inverse is also smooth.
A one parameter diffeomorphism group is a phase flow whose elements are diffeomorphisms satisfying the additional condition that depends smoothly on both and .
A one parameter group of linear transformations is a one parameter diffeomorphism group whose elements are linear transformations.
Every one parameter diffeomorphism group has an associated differential equation.
Every diffeomorphism gives a vector space isomorphism.

Diffeomorphisms as Actions

We can interpret a diffeomorphism as a group action that can act on a variety of objects
The image of the vector field in under a diffeomorphism of a domain onto is the vector field defined by the formula

Where
A diffeomorphism taking a vector field to the field takes the phase curves of the field to the phase curves of the field .
(Arnold 5.4.1) Suppose that there is a direction field defined in the domain . Under the action of a diffeomorphism , the integral curves of the original direction field on map into integral curves of the direction field on obtained by taking the action on the original field.
A diffeomorphism is a symmetry of the vector field on if it maps the field into itself. That is

We say that the vector field is invariant with respect to the symmetry .
- A diffeomorphism is a symmetry of a direction field on if it maps the direction field into itself. The direction field is invariant with respect to the symmetry Integral Curves map onto one another
- A field is invariant with respect to a group of diffeomorphisms if it is invariant with respect to each transformation of the group. We say that the field admits the symmetry group.
- All symmetries of a given field form a group.

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