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