frame

Introduction

Frames and coframes are notions closely related to the notions of basis and dual basis. As such, frames and coframes are needed to describe the connection between list vectors and the more general abstract vectors.

Frames and bases.

Let $\cU$ be a finite-dimensional vector space over a field $\kfield$ , and let $I$ be a finite, totally ordered set of indices ¹, e.g. $(1,2,\ldots,n)$ . We will call a mapping $\bF:I\rightarrow\cU$ a reference frame, or simply a frame. To put it plainly, $\bF$ is just a list of elements of $\cU$ with indices belong to $I$ . We will adopt a notation to reflect this and write $\bF_i$ instead of $\bF(i)$ . Subscripts are used when writing the frame elements because it is best to regard a frame as a row-vector ² whose entries happen to be elements of $\cU$ , and write $$\bF = (\bF_1,\ldots,\bF_n).$$ This is appropriate because every reference frame $\bF$ naturally corresponds to a linear mapping $\hbF:\kfield^I\rightarrow \cU$ defined by $$ \ba \mapsto \sum_{i\in I} \ba^i\, \bF_i,\quad \ba\in \kfield^I.$$ In other words, $\hbF$ is a linear form on $\kfield^I$ that takes values in $\cU$ instead of $\kfield$ . We use row vectors to represent linear forms, and that's why we write the frame as a row vector.

We call $\bF$ a coordinate frame (equivalently, a basis), if $\hbF$ is an isomorphism of vector spaces. Otherwise we call $\bF$ degenerate, incomplete, or both, depending on whether $\hbF$ fails to be, respectively, injective and surjective.

Coframes and coordinates.

In cases where $\bF$ is a basis, the inverse isomorphism $$\hbF^{-1}:\cU\rightarrow \kfield^I$$ is called the coordinate mapping. It is cumbersome to work with this inverse explicitly, and instead we introduce linear forms $\bx^i\in \cU^*,\; i\in I$ defined by $$\bx^i:\bu\mapsto \hbF^{-1}(\bu)(i),\quad \bu\in \cU.$$ Each $\bx^i,\; i\in I$ is called the $i\supth$ coordinate function relative to $\bF$ , or simply the $i\supth$ coordinate ³. In this way we obtain a mapping $$\bx:I\rightarrow \cU^*,\quad i\mapsto \bx^i$$ called the coordinate coframe or simply a coframe. The forms $\bx^i,\,i \in I$ give a basis of $\cU^*$ . It is the dual basis of $\bF_i,\,i \in I$ , i.e. $$\bx^i(\bF_j) = \delta^i_j,\quad i,j\in I,$$ where $\delta^i_j$ is the well-known Kronecker symbol.

In full duality to the custom of writing frames as row-vectors, we write the coframe as a column vector whose components are the coordinate functions: $$ \begin{pmatrix} \bx^1\\ \bx^2\\ \vdots \\ \bx^n \end{pmatrix}. $$ We identify of $\hbF^{-1}$ and $\bx$ with the above column-vector. This is quite natural because all of these objects are in natural correspondence with a $\kfield$ -valued functions of two arguments, $$\cU\times I \rightarrow \kfield,$$ that maps an abstract vector $\bu\in \cU$ and an index $i\in I$ to a scalar $\bx^i(\bu)$ , called the $i\supth$ component of $\bu$ relative to the reference frame $\bF$ .

Change of frame.

Given two coordinate frames $\bF:I\rightarrow \cU$ and $\bG:J\rightarrow \cU$ , one can easily show that $I$ and $J$ must have the same cardinality. Letting $\bx^i,\,i\in I$ and $\by^j,\,j\in J$ denote the coordinates functions relative to $\bF$ and $\bG$ , respectively, we define the transition matrix from $\bF$ to $\bG$ to be the matrix $$\tmat:I\times J\rightarrow \kfield$$ with entries $$\tmat^j_i = \by^j(\bF_i),\quad i\in I,\; j\in J.$$ An equivalent description of the transition matrix is given by $$ \by^j = \sum_{i\in I} \tmat^j_i\, \bx^i,\quad \mbox{for all }j\in J. $$

It is also the custom to regard the elements of $I$ as indexing the columns of the matrix, while the elements of $J$ label the rows. Thus, for $I=(1,2,\ldots,n)$ and $J=(\bar{1},\bar{2},\ldots,\bar{n})$ , we can write $$ \begin{pmatrix} \tmat^{\bar{1}}_1 & \ldots &\tmat^{\bar{1}}_n \\ \vdots & \ddots & \vdots \\ \tmat^{\bar{n}}_1 & \ldots &\tmat^{\bar{n}}_n \end{pmatrix} = \begin{pmatrix} \by^{\bar{1}} \\ \vdots \\ \by^{\bar{n}} \end{pmatrix} \begin{pmatrix} \bF_1 & \ldots & \bF_n \end{pmatrix}. $$ In this way we can describe the relation between coordinates relative to the two frames in terms of ordinary matrix multiplication. To wit, we can write $$ \begin{pmatrix} \by^{\bar{1}} \\ \vdots \\ \by^{\bar{n}} \end{pmatrix} = \begin{pmatrix} \tmat^{\bar{1}}_1 & \ldots &\tmat^{\bar{1}}_n \\ \vdots & \ddots & \vdots \\ \tmat^{\bar{n}}_1 & \ldots &\tmat^{\bar{n}}_n \end{pmatrix} \begin{pmatrix} \bx^1 \\ \vdots \\ \bx^n \end{pmatrix} $$

Notes.

The term frame is often used to refer to objects that should properly be called a moving frame. The latter can be thought of as a field of frames, or functions taking values in the space of all frames, and are fully described elsewhere. The confusion in terminology is unfortunate but quite common, and is related to the questionable practice of using the word scalar when referring to a scalar field (a.k.a. scalar-valued functions) and using the word vector when referring to a vector field.

We also mention that in the world of theoretical physics, the preferred terminology seems to be polyad and related specializations, rather than frame. Most commonly used are dyad, for a frame of two elements, and tetrad for a frame of four elements.

Footnotes

... indices¹

It is advantageous to allow general indexing sets, because one can indicate the use of multiple frames of reference by employing multiple, disjoint sets of indices.

... row-vector²

It is customary to use superscripts for the components of a column vector, and subscripts for the components of a row vector. This is fully described in the vector entry.

... coordinate³

Strictly speaking, we should be denote the coframe by $\bx_\bF$ and the coordinate functions by $\bx^i_\bF$ so as to reflect their dependence on the choice of reference frame. Historically, writers have been loath to do this, preferring a couple of different notational tricks to avoid ambiguity. The cleanest approach is to use different symbols, e.g. $\bx^i$ versus $\by^j$ , to distinguish coordinates coming from different frames. Another approach is to use distinct indexing sets; in this way the indices themselves will indicate the choice of frame. Say we have two frames $\bF: I\rightarrow \cU$ and $\bG:J\rightarrow \cU$ with $I$ and $J$ distinct finite sets. We stipulate that the symbol $i$ refers to elements of $I$ and that $j$ refers to elements of $J$ , and write $\bx^i$ for coordinates relative to $\bF$ and $\bx^j$ for coordinates relative to $\bG$ . That's the way it was done in all the old-time geometry and physics papers, and is still largely the way physicists go about writing coordinates. Be that as it may, the notation has its problems and is the subject of long-standing controversy, named by mathematicians the debauche of indices. The problem is that the notation employs the same symbol, namely $\bx$ , to refer to two different objects, namely a map with domain $I$ and another map with domain $J$ . In practice, ambiguity is avoided because the old-time notation never refers to the coframe (or indeed any tensor) without also writing the indices. This is the classical way of the dummy variable, a cousin to the $f(x)$ notation. It creates some confusion for beginners, but with a little practice it's a perfectly serviceable and useful way to communicate.