Definition Suppose $M$ is a smooth manifold, and $X$ is a smooth vector field on $M$ . Then an integral curve of $X$ through a point $x\in M$ is a curve $c\colon I\to M$ , such that \begin{eqnarray*} c'(t) &=& (X\circ c)(t), \,\,\,\,\,\,\,\mbox{for all $t$ in $I$}\\ c(0) &=& x. \end{eqnarray*}Here $I\subset \sR$ is some open interval of $0$ , and $c'(t)$ is the tangent vector in $T_{c(t)}M$ represented by the curve.

Suppose $x^i$ are local coordinates for $M$ , $c^i$ are functions representing $c$ in these local coordinates, and $X=X^i \frac{\partial}{\partial x^i}$ . Then the condition on $c$ is $$ \frac{dc^i}{dt}(t) = X^i\circ c(t), \quad \mbox{for all $t$}. $$