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Consider the flow (flux) 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. The flow either goes around in circles or goes from sources to sinks. (The sources and sinks might be at Infinity.)

Given a vector field V defined on S, one defines curves γ(t) on S such that for each t in an interval I

$\gamma'(t) = V(\gamma(t))\,.$

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, for some ε > 0,

$\gamma_x(0) = x\,$
$\gamma'_x(t) = V(\gamma_x(t)) \qquad ( t \in (-\varepsilon, +\varepsilon) \subset \mathbf{R}).$

The curves γx are called integral curves or trajectories (or less commonly, flow lines) 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 (i.e., the vector field is equal to the zero vector at the point p), 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.

## Integral curves Three integral curves for the slope field corresponding to the differential equation dy / dx = x2 − x − 2.

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. If the differential equation is represented as a vector field or slope field, then the corresponding integral curves are tangent to the field at each point.

Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In physics, integral curves for an electric field or magnetic field are known as field lines, and integral curves for the velocity field of a fluid are known as streamlines. In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits.

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