PhysicalVector 2002-07-24 07:38:39 2002-10-19 09:47:34 Definition vector \newcommand{\vu}{u} \newcommand{\vv}{v} \newcommand{\cL}{\mathcal{L}} \newcommand{\tmat}{\mathcal{M}} \newcommand{\du}[2]{^{#2}_{\hphantom{#2}\!#1}} \newcommand{\ud}[2]{_{#2}^{\hphantom{#2}\!#1}} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\znums}{\mathbb{Z}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\supth}{^{\text{th}}} \newtheorem{proposition}{Proposition} \newtheorem{definition}[proposition]{Definition} \newcommand{\nl}[1]{{\PMlinkescapetext{{#1}}}} \newcommand{\pln}[2]{{\PMlinkname{{#1}}{#2}}} \paragraph{Definition.} Let $\cL$ be a collection of labels $\alpha\in\cL$. For each ordered pair of labels $(\alpha,\beta)\in\cL\times\cL$ let $\tmat\du{\alpha}{\beta}$ be a non-singular $n\times n$ matrix, the collection of all such satisfying the following functor-like consistency conditions: \begin{itemize} \item For all $\alpha\in \cL$, the matrix $\tmat\du{\alpha}{\alpha}$ is the identity matrix. \item For all $\alpha,\beta,\gamma \in \cL$ we have $$\tmat\du{\alpha}{\gamma} = \tmat\du{\beta}{\gamma}\, \tmat\du{\alpha}{\beta},$$ where the product in the right-hand side is just ordinary matrix multiplication. \end{itemize} We then impose an equivalence relation by stipulating that for all $\alpha,\beta\in\cL$ and $\vu\in\reals^n$, the pair $(\alpha,\vu)$ is equivalent to the pair $(\beta,\tmat\du{\alpha}{\beta}\vu)$. Finally, we define a physical vector to be an equivalence class of such pairs relative to the just-defined relation. The idea behind this definition is that the $\alpha\in \cL$ are labels of various coordinate systems, and that the matrices $\tmat\du{\alpha}{\beta}$ encode the corresponding changes of coordinates. For label $\alpha\in \cL$ and list-vector $\vu\in\reals^n$ we think of the pair $(\alpha,\vu)$ as the representation of a physical vector relative to the coordinate system $\alpha$. \paragraph{Discussion.} All scientific disciplines have a need for formalization. However, the extent to which rigour is pursued varies from one discipline to the next. Physicists and engineers are more likely to regard mathematics as a tool for modeling and prediction. As such they are likely to blur the distinction between list vectors and physical vectors. Consider, for example the following excerpt from R. Feynman's ``Lectures on physics'' \cite{feynman} \begin{quote} All quantities that have a direction, like a step in space, are called vectors. A vector is three numbers. In order to represent a step in space, $\ldots$, we really need three numbers, but we are going to invent a single mathematical symbol, $\mathbf{r}$, which is unlike any other mathematical symbols we have so far used. It is {\em not} a single number, it represents {\em three} numbers: $x$, $y$, and $z$. It means three numbers, but not only {\em those} three numbers, because if we were to use a different coordinate system, the three numbers would be changed to $x'$, $y'$, and $z'$. However, we want to keep our mathematics simple and so we are going to use the {\em same mark} to represent the three numbers $(x,y,z)$ and the three numbers $(x',y',z')$. That is, we use the same mark to represent the first set of three numbers for one coordinate system, but the second set of three numbers if we are using the other coordinate system. This has the advantage that when we change the coordinate system, we do not have to change the letters of our equations. \end{quote} Surely you are joking Mr. Feynman!? What are we supposed to make of this definition? We learn that a vector is both a physical quantity, and a list of numbers. However we also learn that it is not really a specific list of numbers, but rather any of a number of possible lists. Furthermore, the choice of which list is being used is dependent on the context (choice of coordinate system), but this is not really important because we just end up using the same symbol $\mathbf{r}$ regardless. What a muddle! Even at the informal level one can do better than Feynman. The central weakness of his definition is that he is unwilling to distinguish between physical vectors (quantities) and their representation (lists of numbers). Here is an alternative physical definition from a book by R. Aris on fluid mechanics \cite{aris}. \begin{quote} There are many physical quantities with which only a single magnitude can be associated. For example, when suitable units of mass and length have been adopted the density of a fluid may be measured. $\ldots$ There are other quantities associated with a point that have not only a magnitude but also a direction. If a force of 1 lb weight is said to act at a certain point, we can still ask in what direction the force acts and it is not fully specified until this direction is given. Such a physical quantity is a {\em vector}. $\ldots$ We distinguish therefore between the vector as an entity and its components which allow us to reconstruct it {\em in a particular system of reference.} The set of components is meaningless unless the system of reference is also prescribed, just as the magnitude 62.427 is meaningless as a density until the units are also prescribed. $\ldots$. {\em Definition.} A Cartesian vector, $\mathbf{a}$, in three dimensions is a quantity with three components $a_1, a_2, a_3$ in the frame of reference $O123$, which, under rotation of the coordinate frame to $O\bar{1}\bar{2}\bar{3}$, become components $\bar{a}_1, \bar{a}_2, \bar{a}_3$, where $$\bar{a}_j = l_{ij} a_i.$$ The vector $\mathbf{a}$ is to be regarded as an entity, just as the physical quantity it represents is an entity. It is sometimes convenient to use the bold face $\mathbf{a}$ to show this. In any particular coordinate system it has components $a_1, a_2, a_3$, and it is at other times convenient to use the typical component $a_i$. \end{quote} Here we see a carefully drawn distinction between physical quantities and the numerical measurements that represent them. A system of measurement, i.e. a choice of units and or coordinate axes, turns physical quantities into numbers. However the correspondence is not fixed, but varies according to the choice of measurement system. This point of view can be formalized by representing physical vectors as labeled list vectors, the label specifying a choice of measurement systems. The actual vector is then defined to be an equivalence class of such labeled list vectors. \begin{thebibliography}{99} \bibitem{feynman} R. Feynman, R. Leighton, and M. Sands, ``Lectures on Physics'', 11-4, Vol. I, Addison-Wesley. \bibitem{aris} R. Aris, ``Vectors, Tensors and the Basic Equations of Fluid Mechanics'', Dover. \end{thebibliography}