Virtausjoukko

Virtausjoukko is a term used in mathematics and physics, particularly in the study of dynamical systems and fluid mechanics. It refers to the time evolution of points in a space under a continuous flow, described by a family of maps φt: M → M parameterized by real time t. The flow represents how a point moves under a given velocity field or dynamical rule.

In formal terms, the virtausjoukko is the flow of a vector field X on a manifold M. The maps satisfy φ0 = idM and φt+s = φt ∘ φs for all t, s in the domain. The vector field X is the infinitesimal generator of the flow, meaning that for each point p in M, the trajectory t ↦ φt(p) solves the ordinary differential equation d/dt φt(p) = X(φt(p)) with φ0(p) = p. If X is complete, the flow is defined for all times t in R and consists of a one-parameter group of diffeomorphisms of M.

Virtausjoukko is used to describe particle motion in a fluid, where each fluid particle follows a trajectory t → φt(p) determined by the velocity field. It also appears in numerical and theoretical contexts, such as advection problems, optical flow, and the study of dynamical systems through flow maps.

Related terms include the flow map, one-parameter Lie group of diffeomorphisms, and the distinction between Eulerian descriptions (fields) and Lagrangian descriptions (particle paths).