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.