digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

Three integral curves for the slope field corresponding to the differential equation dy / dx = x2 − x − 1.

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,...,Fn); and x(t) 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 following autonomous system of ordinary differential equations:

\begin{align} \frac{dx_1}{dt} &= F_1(x_1,\ldots,x_n) \\ &\vdots \\ \frac{dx_n}{dt} &= F_n(x_1,\ldots,x_n). \end{align}

Such a system may be written as a single vector equation

$\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)).\!\,$

This equation says precisely that the tangent vector 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.

Generalization to differentiable manifolds

Definition

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 : TMM given by

$\pi_{M} : (x, v) \mapsto x.$

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 pM. An integral curve for X passing through p at time t0 is a curve α : JM of class Cr−1, defined on an open interval J of the real line R containing t0, such that

$\alpha (t_{0}) = p;\,$
$\alpha' (t) = X (\alpha (t)) \mbox{ for all } t \in J.$

Relationship to ordinary differential equations

The above definition of an integral curve α for a vector field X, passing through p at time t0, is the same as saying that α is a local solution to the ordinary differential equation/initial value problem

$\alpha (t_{0}) = p;\,$
$\alpha' (t) = X (\alpha (t)).\,$

It is local in the sense that it is defined only for times in J, and not necessarily for all tt0 (let alone tt0). Thus, the problem of proving the existence and uniqueness of integral curves is the same as that of finding solutions to ordinary differential equations/initial value problems and showing that they are unique.

Remarks on the time derivative

In the above, α′(t) denotes the derivative of α at time t, the "direction α is pointing" at time t. From a more abstract viewpoint, this is the Fréchet derivative:

$(\mathrm{d}_t f) (+1) \in \mathrm{T}_{\alpha (t)} M.$

In the special case that M is some open subset of Rn, this is the familiar derivative

$\left( \frac{\mathrm{d} \alpha_{1}}{\mathrm{d} t}, \dots, \frac{\mathrm{d} \alpha_{n}}{\mathrm{d} t} \right),$

where α1, ..., αn are the coordinates for α with respect to the usual coordinate directions.

The same thing may be phrased even more abstractly in terms of induced maps. Note that the tangent bundle TJ of J is the trivial bundle J × R and there is a canonical cross-section ι of this bundle such that ι(t) = 1 (or, more precisely, (t, 1)) for all tJ. The curve α induces a bundle map α : TJ → TM so that the following diagram commutes:

Then the time derivative α′ is the composition α′ = α o ι, and α′(t) is its value at some point t ∈ J.

References

• Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Integral_curve — Please support Wikipedia.
 512442 videos foundNext >
 Lec 1 | MIT 18.03 Differential Equations, Spring 2006The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves. View the complete course: http://ocw.mit.edu/18-03S06 License: Creative Commons BY-NC-S... Line Integrals - Evaluating a Line IntegralLine Integrals - Evaluating a Line Integral - I give the basic formula and do one example of evaluating a line integral. For more free math videos, visit htt... Introduction to the Line IntegralMore free lessons at: http://www.khanacademy.org/video?v=_60sKaoRmhU Introduction to the Line Integral. Line Integral Example 1More free lessons at: http://www.khanacademy.org/video?v=uXjQ8yc9Pdg Concrete example using a line integral. Area Under a Curve: Introduction to Integral CalculusIn this video I discuss what the area under a curve means and show how you can sum up simple rectangle shapes and take the limit of them toward to infinite a... Line integral from vector calculus over a closed curveFree ebook http://tinyurl.com/EngMathYT I present an example where I calculate the line integral of a given vector function over a closed curve.. In particul... Integral Calculus lesson 4 - How to find the area under a curve (definite integrals)In this lesson we learn how integrating an equation will give us an equation for calculating the area under the curve. We then use definite integrals to find... Area Between Curves - Integrating with Respect to yArea Between Curves - Integrating with Respect to y - I show the general formula, graph two functions, and set up the integral in this video. I calculate the... Introduction to definite integralsMore free lessons at: http://www.khanacademy.org/video?v=0RdI3-8G4Fs Using the definite integral to solve for the area under a curve. Intuition on why the an... Integral Calculus lesson 6 - Area bounded by the curve when it cuts through the x axisIn this lesson we talk about how you can find the area between the curve and the x axis if the curve goes through the x axis.
 512442 videos foundNext >

We're sorry, but there's no news about "Integral curve" right now.

 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter