integral surface

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.

Name

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 is a static vector field, that is, a vector-valued function with Cartesian coordinates , and that is a parametric curve with Cartesian coordinates . Then is an integral curve of if it is a solution of the autonomous system of ordinary differential equations,

$\begin \frac &= F_1(x_1,\ldots,x_n) \\ &\;\, \vdots \\ \frac &= F_n(x_1,\ldots,x_n). \end$

Such a system may be written as a single vector equation,

$\mathbf'(t) = \mathbf(\mathbf(t)).$

This equation says that the vector tangent to the curve at any point along the curve is precisely the vector , and so the curve 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.

Examples

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.

Generalization to differentiable manifolds

Definition

Let be a Banach manifold of class with . As usual, denotes the tangent bundle of with its natural projection given by

$\pi_M : (x, v) \mapsto x.$

A vector field on is a cross-section of the tangent bundle , i.e. an assignment to every point of the manifold of a tangent vector to at that point. Let be a vector field on of class and let . An integral curve for passing through at time is a curve of class , defined on an open interval of the real line containing , such that

$\begin \alpha(t_0) &= p; \\ \alpha' (t) &= X (\alpha (t)) \text t \in J. \end$

Relationship to ordinary differential equations

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

$\begin \alpha(t_0) &= p; \\ \alpha' (t) &= X (\alpha (t)). \end$

It is local in the sense that it is defined only for times in , and not necessarily for all (let alone ). 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, denotes the derivative of at time , the "direction is pointing" at time . From a more abstract viewpoint, this is the Fréchet derivative:

$(\mathrm_t\alpha) (+1) \in \mathrm_ M.$

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

$\left( \frac, \dots, \frac \right),$

where are the coordinates for 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 of is the trivial bundle and there is a canonical cross-section of this bundle such that (or, more precisely, ) for all . The curve induces a bundle map so that the following diagram commutes:

:

Then the time derivative is the composition is its value at some point .

References

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{{Manifolds

Differential geometry
Ordinary differential equations