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
It is customary to use the short notation . Each velocity component can be shortened to . Using to denote the vector of functions , , , (8.11) can be shorted to
The differential equations expressed in (8.11) are often referred to as autonomous or stationary because does not depend on time. A time-varying vector field could alternatively be defined, which yields . This will not be covered, however, in this chapter.
Steven M LaValle 2012-04-20