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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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