integralkurvan

Integralkurvan, in Swedish mathematical terminology, refers to the concept of an integral curve. In differential geometry and the theory of ordinary differential equations, an integral curve is a curve that follows the direction specified by a vector field or by a first-order system of differential equations. More concretely, if X is a smooth vector field on a differentiable manifold M, an integral curve is a map γ from an interval I ⊆ R to M such that γ′(t) = X(γ(t)) for all t in I. In R^n this becomes an ordinary differential equation x′(t) = F(x(t)) with γ(0) = x0, where F is the vector field represented as a function from R^n to R^n.

For a given initial point p ∈ M, the integral curve through p is the solution γp(t) of the differential equation with γp(0) = p. The collection of integral curves forms the flow of the vector field: a family of maps φt: U → M, where φt(p) = γp(t) and U is the largest domain on which the solution exists. When the system is autonomous, these curves are also called trajectories or orbits of the system.

Existence and uniqueness theorems provide the conditions under which integral curves exist and are unique. If the vector field is locally Lipschitz, a unique local integral curve passes through each point; continuity alone guarantees local existence (Peano’s theorem) but not uniqueness.

Examples include linear systems x′ = Ax in R^n, whose integral curves are given by x(t) = e^{tA}x0, and planar systems where the slope dy/dx equals the ratio of the vector field components along a curve. Integralkurvan thus captures the geometric flow determined by a differential equation.