The steady flow of a fluid in a Euclidean space is an appropriate model for the ideas developed in this chapter. The essential ideas are
(1) that the fluid is supposed to fill the space, so that there is a streamline through each point
(2) that the velocity of the fluid at each point specifies a vector field in the space
(3) that the movement of the fluid along the streamlines for a fixed interval of time specifies a transformation of the space into itself.
The fluid flow is thus considered both passively, as a collection of streamlines, and actively, as a collection of transformations of the space. Besides these integral appearances it also appears differentially, through its velocity field.
Let ϕt denote the transformation of the space into itself by movement along the streamlines during a time interval of length t. To be specific, given any point x of the space, ϕt(x) is the point reached by a particle of the fluid, initially at x and flowing along the streamline of the fluid through x, after the lapse of a time t. The set of such transformations has the almost self-evident properties
(1) ϕ0 is the identity transformation
(2) ϕs ∘ ϕt = ϕs+t.
A set of transformations with these two properties (for all s and t) is called a one-parameter group of transformations. The study of such transformations, and of the streamlines and vector fields associated with them, forms the subject matter of this chapter.