integralkurva

In Swedish mathematical terminology, integralkurva denotes an integral curve. An integral curve of a vector field F on a smooth manifold M is a differentiable curve γ: I → M such that γ′(t) = F(γ(t)) for all t in I. In Euclidean space R^n, with F: R^n → R^n, this means γ solves the ordinary differential equation dγ/dt = F(γ).

Given an initial time t0 and an initial point x0 ∈ M, the initial value problem dγ/dt = F(γ), γ(t0) = x0 seeks an integral curve passing through x0 at t0. If F is Lipschitz continuous (for example, C^1), there exists a unique maximal integral curve γ with γ(t0) = x0. The time-t flow φ_t defined by φ_t(x0) = γ(t0 + t) describes how points move under the system and is called the flow of F.

Integral curves are tangent to the vector field at every point and, collectively, foliate the domain where the flow is defined. Special cases include constant vector fields, where integral curves are stationary, and linear vector fields F(x) = Ax, whose integral curves have the form γ(t) = e^{tA}x0.

Examples help illustrate the concept. A planar system dx/dt = y, dy/dt = −x yields integral curves that are circles centered at the origin, corresponding to conservative motion. A system dx/dt = x, dy/dt = y produces trajectories γ(t) = (x0 e^t, y0 e^t), which are exponential routes along straight lines from the origin.

Integral curves are also known as solution curves or trajectories and are central to dynamical systems, differential geometry, and physics for analyzing stability, phase portraits, and long-term behavior.