These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory.
Every smooth vector field on a compact manifold without boundary is complete.
Splitting the tangent vectors into directional derivatives, one can solve the resulting system of differential equations to find the function.
From a point, the rate of change of the i-th component with respect to the parametrization of the flow (“how much the flow has acted”) is described by the i-th component of the field.
Every left-invariant vector field on a Lie group is complete.