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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Smooth manifolds constitute a certain class of topological spaces which locally look like some Euclidean space R^n and on which one can do calculus. We introduce the key concepts of this subject, such
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.
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Explores curve integrals of vector fields, energy calculations, potential functions, and tangential vectors, with a focus on line integrals and domains.
Explores integral curves on manifolds and the significance of the exponential map in Lie groups.
Explores the application of the Gauss/Green theorem to calculate curve integrals along simple closed curves.
We construct divergence-free Sobolev vector fields in C([0,1];W-1,W-r(T-d;Rd)) with r < d and d\geq 2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. These vector fields are then shown to have at leas ...
Two Lie algebroids are presented that are linked to the construction of the linearizing output of an affine in the input nonlinear system.\ The algorithmic construction of the linearizing output proceeds inductively, and each stage has two structures, name ...
The Gröbner walk is an algorithm for conversion between Gröbner bases for different term orders. It is based on the polyhedral geometry of the Gröbner fan and involves tracking a line between cones representing the initial and target term order. An importa ...