Integrallinien

Integrallinien are curves in a domain that are tangent at every point to a given vector field. In English, the same concept is usually called integral curves of a vector field F: R^n → R^n. Equivalently, they are the trajectories obtained by solving the differential equation x'(t) = F(x(t)) for autonomous systems, or x'(t) = F(x(t), t) for non-autonomous systems.

Given an initial point x0 in the domain, the integral line through x0 is the solution x(t) with x(0) = x0, defined on a maximal interval about t = 0. If the vector field F is sufficiently smooth—for example Lipschitz continuous on a region—existence and uniqueness theorems guarantee a unique integral line locally, and often globally, depending on the growth of F.

Geometrically, integral lines describe the flow generated by the vector field. They partition the space into orbits along which a particle would move if its velocity at each point is given by F. In two dimensions they are often used to produce phase portraits that visualize system behavior and stability properties.

In fluid dynamics, related concepts include streamlines, pathlines and streaklines. For steady flows, integral lines of the velocity field coincide with streamlines; for time-dependent flows they generally do not, reflecting the difference between instantaneous direction and actual particle paths.

Computation and applications: When F is simple, integral lines can be found analytically by solving the differential equations. In most cases numerical methods such as Euler or higher-order Runge-Kutta schemes are used to integrate initial value problems. Integral lines are central to the study of dynamical systems, physics, biology, and engineering, where they reveal invariant sets, stability, and long-term behavior.