The notation for a tangent space on a manifold looks the same as for . This enables the vector field definition and notation to extend naturally from to smooth manifolds. A vector field on a manifold assigns a vector in for every . It can once again be imagined as a needle diagram, but now the needle diagram is spread over the manifold, rather than lying in .
The velocity field interpretation of a vector field can also be extended to smooth manifolds. This means that now defines a set of differential equations over and is usually expressed using a coordinate neighborhood of the smooth structure. If is a smooth vector field, then a solution trajectory, , can be defined from any . Solution trajectories in the sense of Filipov can also be defined, for the case of piecewise-smooth vector fields.
Steven M LaValle 2012-04-20