Editing Integral curve (section)

===Relationship to ordinary differential equations===

The above definition of an integral curve {{math|''α''}} for a vector field {{math|''X''}}, passing through {{math|''p''}} at time {{math|''t''<sub>0</sub>}}, is the same as saying that {{math|''α''}} is a local solution to the ordinary differential equation/initial value problem

<math display="block">\begin{align}
\alpha(t_0) &= p; \\
\alpha' (t) &= X (\alpha (t)).
\end{align}</math>

It is local in the sense that it is defined only for times in {{math|''J''}}, and not necessarily for all {{math|''t'' ≥ ''t''<sub>0</sub>}} (let alone {{math|''t'' ≤ ''t''<sub>0</sub>}}). 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.