Flow of a vector field

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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 .


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 .


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


for everyone .


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.


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