Vector fields as velocity fields

We now give a particular interpretation to vector fields. A vector field expressed using (8.10) can be used to define a set of first-order differential equations as

$$\begin{split}\frac{dx_1}{dt} = f_1(x_1,\ldots,x_n) \frac{d... ...uein} \frac{dx_n}{dt} = f_n(x_1,\ldots,x_n) . \end{split}$$ (8.11)

Each equation represents the derivative of one coordinate with respect to time. For any point $x \in {\mathbb{R}}^n$, a velocity vector is defined as

$$\frac{dx}{dt} = \left[\frac{dx_1}{dt} \;\; \frac{dx_2}{dt} \;\; \cdots \;\; \frac{dx_n}{dt} \right].$$ (8.12)

This enables $f$ to be interpreted as a velocity field.

It is customary to use the short notation ${\dot x}= dx/dt$. Each velocity component can be shortened to ${\dot x}_i = dx_i/dt$. Using $f$ to denote the vector of functions $f_1$, $\ldots$, $f_n$, (8.11) can be shorted to

$${\dot x}= f(x) .$$ (8.13)

The use of $f$ here is an intentional coincidence with the use of $f$ for the state transition equation. In Part IV, we will allow vector fields to be parameterized by actions. This leads to a continuous-time state transition equation that looks like ${\dot x}= f(x,u)$ and is very similar to the transition equations defined over discrete stages in Chapter 2.

The differential equations expressed in (8.11) are often referred to as autonomous or stationary because $f$ does not depend on time. A time-varying vector field could alternatively be defined, which yields ${\dot x}= f(x(t),t)$. This will not be covered, however, in this chapter.