Vector fields and velocity fields on manifolds

The notation for a tangent space on a manifold looks the same as for $ {\mathbb{R}}^n$. This enables the vector field definition and notation to extend naturally from $ {\mathbb{R}}^n$ to smooth manifolds. A vector field on a manifold $ M$ assigns a vector in $ T_p(M)$ for every $ p \in M$. It can once again be imagined as a needle diagram, but now the needle diagram is spread over the manifold, rather than lying in $ {\mathbb{R}}^n$.

The velocity field interpretation of a vector field can also be extended to smooth manifolds. This means that $ {\dot x}= f(x)$ now defines a set of $ n$ differential equations over $ M$ and is usually expressed using a coordinate neighborhood of the smooth structure. If $ f$ is a smooth vector field, then a solution trajectory, $ \tau
: [0,\infty) \rightarrow M$, can be defined from any $ x_0 \in M$. 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