Integral Curve - Definition

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

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.

Read more about this topic:  Integral Curve

Famous quotes containing the word definition:

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)