Flow Curves
Consider the flow of a fluid through a region of space. At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow. The converse is also true: it is possible to associate a flow to a vector field having that vector field as its velocity.
Given a vector field V defined on S, one defines curves γ(t) on S such that for each t in an interval I
By the Picard–Lindelöf theorem, if V is Lipschitz continuous there is a unique C1-curve γx for each point x in S so that
The curves γx are called flow curves of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval (−ε, +ε) to the whole real number line. The flow may for example reach the edge of S in a finite time. In two or three dimensions one can visualize the vector field as giving rise to a flow on S. If we drop a particle into this flow at a point p it will move along the curve γp in the flow depending on the initial point p. If p is a stationary point of V then the particle will remain at p.
Typical applications are streamline in fluid, geodesic flow, and one-parameter subgroups and the exponential map in Lie groups.
Read more about this topic: Vector Field
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