# Flow of a vector field

In mathematics , the **flow of a vector field describes** the movement along the solution curves of the ordinary differential equation given by the vector field .

## definition

Let be a - vector field on an open subset (or more generally on an open subset of a manifold ). According to the existence and uniqueness theorem for ordinary differential equations, there is a unique maximum solution of the differential equation for each

- .

Here is the (possibly infinite) maximum interval on which a solution is defined.

Be . Then it's called through

given illustration the *flow of the vector field* .

## properties

The flow of a vector field is a flow , i. H. a one-parameter transformation group. So it applies

and

for everyone .

## example

The flow of the vector field defined on the is given by .

## Complete vector fields

The vector field is called a **complete vector field** if its flow is defined for all times, i.e. for all , or is equivalent .

Vector fields with compact carriers are always complete. This is especially true for vector fields on compact manifolds.

## literature

- John Lee: "Introduction to smooth manifolds", Graduate Texts in Mathematics, Springer, ISBN 978-0-387-21752-9
- Vladimir Arnold: "Ordinary differential equations", Universitext, Springer, ISBN 978-3-540-34563-3

## Web links

- Flow of a Vectorfield (nLab)