 Given a vector field "V" defined on "S", one defines curves γ("t") on "S" such that for each "t" in an interval "I"

By the Picard–Lindelöf theorem, if "V" is Lipschitz continuous there is a "unique" "C"1-curve γ"x" for each point "x" in "S" so that, for some ε > 0,

The curves γ"x" are called integral curves or trajectories (or less commonly, flow lines) of the vector field "V" and partition "S" into equivalence classes.It is not always possible to extend the interval (−ε, +ε) to the whole real number line.The flow may for example reach the edge of "S" in a finite time.