In mathematics, flow refers to the group action of a one-parameter group on a set. Flows typically arise as the solutions of ordinary differential equations. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, and the Anosov flow. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... This article is about the mathematical concept. ... In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ... In differential geometry, Ricci flow is the flow of Riemannian metrics given by the equation where g is the metric and Ric is the Ricci curvature. ... In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of expansion and contraction. Anosov diffeomorphisms were introduced by D. V. Anosov, who proved...

Formal definition

A flow on a set X is a group action of on X. More explicitly, a flow is a function with and that is consistent with the structure of a one-parameter group: This article is about the mathematical concept. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective...

for all s,t in and

The set is called the orbit of x by In mathematics, groups are often used to describe symmetries of objects. ...

Flows are usually required to be continuous or even differentiable, when the space X has some additional structure (e.g. when X is a topological space or when ) In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

The most common examples of flows arise from describing the solutions of the autonomous ordinary differential equation In differential equations, an autonomous system is an equation of the form where x takes values in n-dimensional Euclidean space and t is usually time. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. ...

as a function of the initial condition x, when the equation has existence and uniqueness of solutions. That is, if the equation above has a unique solution for each , then defines a flow. In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. ...

This article incorporates material from Flow on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...