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Concept
Integral curve
Summary
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
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.
Suppose that F is a static vector field, that is, a vector-valued function with Cartesian coordinates (F1,F2,...,Fn), and that x(t) is a parametric curve with Cartesian coordinates (x1(t),x2(t),...,xn(t)). Then x(t) is an integral curve of F if it is a solution of the autonomous system of ordinary differential equations,
Such a system may be written as a single vector equation,
This equation says that the vector tangent to the curve at any point x(t) along the curve is precisely the vector F(x(t)), and so the curve x(t) is tangent at each point to the vector field F.
If a given vector field is Lipschitz continuous, then the Picard–Lindelöf theorem implies that there exists a unique flow for small time.
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.
Let M be a Banach manifold of class Cr with r ≥ 2. As usual, TM denotes the tangent bundle of M with its natural projection πM : TM → M given by
A vector field on M is a cross-section of the tangent bundle TM, i.e. an assignment to every point of the manifold M of a tangent vector to M at that point. Let X be a vector field on M of class Cr−1 and let p ∈ M.
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Related publications (4)
Related concepts (3)
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups.
Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.
Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.