# 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${\ displaystyle F}$${\ displaystyle C ^ {1}}$${\ displaystyle U \ subset \ mathbb {R} ^ {n}}$${\ displaystyle x_ {0} \ in U}$ ${\ displaystyle \ gamma _ {x_ {0}} \ colon (a (x_ {0}), b (x_ {0})) \ to \ mathbb {R}}$

${\ displaystyle {\ dot {x}} = F (x), x (0) = x_ {0}}$.

Here is the (possibly infinite) maximum interval on which a solution is defined. ${\ displaystyle (a (x_ {0}), b (x_ {0}))}$

Be . Then it's called through ${\ displaystyle \ Sigma _ {F} = \ left \ {(t, x) \ in \ mathbb {R} \ times U \ colon a (x)

${\ displaystyle \ Phi (t, x): = \ gamma _ {x} (t)}$

given illustration the flow of the vector field . ${\ displaystyle \ Phi \ colon \ Sigma _ {F} \ to U}$ ${\ displaystyle F}$

## properties

Vector field F (x, y) = (- y, x)

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

${\ displaystyle \ Phi (0, x) = x}$

and

${\ displaystyle \ Phi (s + t, x) = \ Phi (s, \ Phi (t, x))}$

for everyone . ${\ displaystyle x \ in U}$

## example

The flow of the vector field defined on the is given by . ${\ displaystyle \ mathbb {R} ^ {2}}$${\ displaystyle F (x, y) = (- y, x)}$${\ displaystyle \ Phi (t, (x, y)) = (\ cos (t) x + \ sin (t) y, - \ sin (t) x + \ cos (t) y)}$

## 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 . ${\ displaystyle F}$${\ displaystyle a (x_ {0}) = - \ infty, b (x_ {0}) = \ infty}$${\ displaystyle x_ {0} \ in U}$${\ displaystyle \ Sigma _ {F} = \ mathbb {R} \ times U}$

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