Vector fields

A vector field, ${\vec V}$, on a manifold X, is a function that associates with each $x \in X$, a vector, ${\vec V}(x)$. The velocity field is a special vector field that will be used extensively. Each vector ${\vec V}(x)$ in a velocity field represents the infinitesimal change in state with respect to time,  

$${\dot x}= \left[ \frac{dx_1}{dt} \;\; \frac{dx_2}{dt} \; \cdots \; \frac{dx_n}{dt} \right] ,$$ (2)

evaluated at the point $x \in X$.

Note that for a fixed u, any state transition equation, ${\dot x}= f(x,u)$ defines a vector field because ${\dot x}$ is expressed as a function of x.