Flow (mathematics)

A family of transformations obtained by evolving points along a vector field for a specified amount of time. Each transformation advances every point according to the same differential equation.

Ordinary differential equation: An equation relating an unknown function to its derivatives with respect to one or more variables. A flow is generated by solving an ordinary differential equation from each initial point.
Differentiable manifold: A space that locally resembles Euclidean space and supports consistently defined derivatives. Flows commonly evolve points on manifolds, including spaces with curved geometry.
Dynamical system: A mathematical model describing how a system's state changes over time according to a rule. A flow is the continuous-time form of a deterministic dynamical system.
Discrete dynamical system: A system whose state advances through repeated applications of a map indexed by integer time. Unlike a flow, it need not provide transformations for every real elapsed time.
Vector field: A field assigning a vector to each point of a space, often specifying a direction and rate of motion. The vector field supplies the instantaneous velocity that generates the flow.
Tangent space: A vector space representing the possible instantaneous directions of motion at a point on a manifold. A vector field's value at each point belongs to that point's tangent space.
Hamiltonian mechanics: A formulation of classical mechanics in which motion is generated by a Hamiltonian function on phase space. Hamiltonian functions generate flows that describe conservative mechanical motion.
Semiflow: A family of time-evolution maps defined for nonnegative times and satisfying the composition law. A semiflow permits forward evolution without requiring inverse maps or negative-time solutions.
Flow map: A map sending each initial point to its position after evolution for a fixed time. Each fixed-time transformation in a flow is a flow map.
Integral curve: A curve whose velocity at every point equals the value of a specified vector field. A flow traces an integral curve from each initial point.

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