Integralkurve

An Integralkurve, in mathematics commonly called an integral curve, is a curve that follows the direction given by a vector field. Formally, let M be a smooth manifold and X a smooth vector field on M. An integral curve is a differentiable map γ: I → M defined on an interval I such that γ'(t) = X(γ(t)) for every t in I. If a point p ∈ M satisfies γ(t0) = p for some t0 ∈ I, then γ is the trajectory of the vector field through p. In Euclidean space R^n, this means solving a system of ordinary differential equations dx/dt = F(x), and the integral curves are the solution curves through a specified initial condition x(t0) = x0.

Existence and uniqueness results, such as the Picard–Lindelöf theorem, guarantee a unique local integral curve through a given point when the vector field satisfies appropriate regularity conditions (for example, being Lipschitz in a neighborhood). Global existence may fail if solutions blow up in finite time or leave the domain.

Examples clarify the concept: if X is a constant vector field a, then integral curves are straight lines γ(t) = p + t a. For a nonlinear field in two dimensions, integral curves satisfy dy/dx = g(x,y)/f(x,y) wherever f ≠ 0, tracing out the slope field defined by the vector field.

Relation to dynamical systems: integral curves are the orbits of the flow generated by X, describing system trajectories and forming the basis of phase portraits and stability analysis.