Integral curve

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.

Definition

Suppose that F is a vector field: that is, a vector-valued function with Cartesian coordinates (F₁,F₂,...,F_n); and x(t) a parametric curve with Cartesian coordinates (x₁(t),x₂(t),...,x_n(t)). Then x(t) is an integral curve of F if it is a solution of the following autonomous system of ordinary differential equations: