A vector field looks like a ``needle diagram'' over , as depicted in Figure 8.5. The idea is to specify a direction at each point . When used to represent a feedback plan, it indicates the direction that the robot needs to move if it finds itself at .

For every
, associate an -dimensional vector space
called the *tangent space* at , which is denoted as
. Why not just call it a vector space at ? The use of
the word ``tangent'' here might seem odd; it is motivated by the
generalization to manifolds, for which the tangent spaces will be
``tangent'' to points on the manifold.

A *vector field*^{8.4} on
is a function that
assigns a vector
to every
. What is
the range of this function? The vector
at each
actually belongs to a different tangent space. The range of the
function is therefore the union

which is called the

A vector field can therefore be expressed using real-valued functions on . Let for from to denote such functions. Using these, a vector field is specified as

In this case, it appears that a vector field is a function from into . Therefore, standard function notation will be used from this point onward to denote a vector field.

Now consider some examples of vector fields over . Let a point in be represented as . In standard vector calculus, a vector field is often specified as , in which and are functions on

Due to obstacles that arise in planning problems, it will be convenient to sometimes restrict the domain of a vector field to an open subset of . Thus, for any open subset , a vector field can be defined.

Steven M LaValle 2012-04-20