Vector fields

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

For every $ p \in {\mathbb{R}}^n$, associate an $ n$-dimensional vector space called the tangent space at $ p$, which is denoted as $ T_p({\mathbb{R}}^n)$. Why not just call it a vector space at $ p$? 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 field8.4 $ {\vec V}$ on $ {\mathbb{R}}^n$ is a function that assigns a vector $ v \in T_p({\mathbb{R}}^n)$ to every $ p \in {\mathbb{R}}^n$. What is the range of this function? The vector $ {\vec V}(p)$ at each $ p \in {\mathbb{R}}^n$ actually belongs to a different tangent space. The range of the function is therefore the union

$\displaystyle T({\mathbb{R}}^n) = \bigcup_{p \in {\mathbb{R}}^n} T_p({\mathbb{R}}^n) ,$ (8.9)

which is called the tangent bundle on $ {\mathbb{R}}^n$. Even though the way we describe vectors from $ T_p({\mathbb{R}}^n)$ may appear the same for any $ p \in {\mathbb{R}}^n$, each tangent space is assumed to produce distinct vectors. To maintain distinctness, a point in the tangent bundle can be expressed with $ 2n$ coordinates, by specifying $ p$ and $ v$ together. This will become important for defining phase space concepts in Part IV. In the present setting, it is sufficient to think of the range of $ {\vec V}$ as $ {\mathbb{R}}^n$ because $ T_p({\mathbb{R}}^n) = {\mathbb{R}}^n$ for every $ p \in {\mathbb{R}}^n$.

A vector field can therefore be expressed using $ n$ real-valued functions on $ {\mathbb{R}}^n$. Let $ f_i(x_1,\ldots,x_n)$ for $ i$ from $ 1$ to $ n$ denote such functions. Using these, a vector field is specified as

$\displaystyle f(x) = [f_1(x_1,\ldots,x_n) \;\; f_2(x_1,\ldots,x_n) \;\; \cdots \;\; f_n(x_1,\ldots,x_n)] .$ (8.10)

In this case, it appears that a vector field is a function $ f$ from $ {\mathbb{R}}^n$ into $ {\mathbb{R}}^n$. Therefore, standard function notation will be used from this point onward to denote a vector field.

Now consider some examples of vector fields over $ {\mathbb{R}}^2$. Let a point in $ {\mathbb{R}}^2$ be represented as $ p = (x,y)$. In standard vector calculus, a vector field is often specified as $ [f_1(x,y) \;\;\;
f_2(x,y)]$, in which $ f_1$ and $ f_2$ are functions on $ {\mathbb{R}}^2$

Figure: (a) A constant vector field, $ f(x,y) =
[1\;\;1]$. (b) A vector field, $ f(x,y) = [-x\;\;-y]$ in which all vectors point to the origin.
\psfig{file=figs/constant.eps,w...,width=2.7in} \\
(a) & (b)

Example 8..6 (Constant Vector Field)   Figure 8.5a shows a constant vector field, which assigns the vector $ [1 \;\; 2]$ to every $ (x,y) \in {\mathbb{R}}^2$. $ \blacksquare$

Example 8..7 (Inward Flow)   Figure 8.5b depicts a vector field that assigns $ [-x
\;\;\; -y]$ to every $ (x,y) \in {\mathbb{R}}^2$. This causes all vectors to point to the origin. $ \blacksquare$

Figure: A swirling vector field, $ f(x,y) =
\begin{figure}\centerline{\psfig{file=figs/swirl.eps,width=3.0truein,height=3.0truein} }\end{figure}

Example 8..8 (Swirl)   The vector field in Figure 8.6 assigns $ [(y-x) \;\;\; (-x-y)]$ to every $ (x,y) \in {\mathbb{R}}^2$. $ \blacksquare$

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 $ {\mathbb{R}}^n$. Thus, for any open subset $ O \subset {\mathbb{R}}^n$, a vector field $ f : O \rightarrow {\mathbb{R}}^n$ can be defined.

Steven M LaValle 2020-08-14