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.

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