Editing Integral curve (section)

==Definition==
Suppose that {{math|'''F'''}} is a static [[vector field]], that is, a [[vector-valued function]] with [[Cartesian coordinate system|Cartesian coordinates]] {{math|(''F''<sub>1</sub>,''F''<sub>2</sub>,...,''F''<sub>''n''</sub>)}}, and that {{math|'''x'''(''t'')}} is a [[parametric curve]] with Cartesian coordinates {{math|(''x''<sub>1</sub>(''t''),''x''<sub>2</sub>(''t''),...,''x''<sub>''n''</sub>(''t''))}}. Then {{math|'''x'''(''t'')}} is an '''integral curve''' of {{math|'''F'''}} if it is a solution of the [[autonomous system (mathematics)|autonomous system]] of ordinary differential equations,

<math display="block">\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}
</math>

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

<math display="block">\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)).</math>

This equation says that the vector tangent to the curve at any point {{math|'''x'''(''t'')}} along the curve is precisely the vector {{math|'''F'''('''x'''(''t''))}}, and so the curve {{math|'''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.