frame

Introduction

Frames and coframes are notions closely related to the notions ofbasis and dual basis. As such, frames and coframes are needed todescribe the connection between list vectors (http://planetmath.org/Vector2) andthe more general abstract vectors (http://planetmath.org/VectorSpace).

Frames and bases.

Let $\\mathcal{U}$ be a finite-dimensional vector space over a field $\\mathbb{K}$ ,and let $I$ be a finite, totally ordered set of indices¹¹It isadvantageous to allow general indexing sets, because one canindicate the use of multiple frames of reference by employingmultiple, disjoint sets of indices., e.g. $(1,2,\\ldots,n)$ . Wewill call a mapping $\\mathbf{F}:I\\rightarrow\\mathcal{U}$ a reference frame, or simplya frame. To put it plainly, $\\mathbf{F}$ is just a list of elements of $\\mathcal{U}$ with indices belong to $I$ . We will adopt a notation to reflect thisand write $\\mathbf{F}_{i}$ instead of $\\mathbf{F}(i)$ . Subscripts are used whenwriting the frame elements because it is best to regard a frame as arow-vector²²It is customary to use superscripts for thecomponents of a column vector, and subscripts for the components ofa row vector. This is fully described in thevector entry (http://planetmath.org/Vector2). whose entries happen to be elements of $\\mathcal{U}$ ,and write

\\mathbf{F}=(\\mathbf{F}_{1},\\ldots,\\mathbf{F}_{n}).

This is appropriate becauseevery reference frame $\\mathbf{F}$ naturally corresponds to a linearmapping $\\hat{\\mathbf{F}}:\\mathbb{K}^{I}\\rightarrow\\mathcal{U}$ defined by

\\mathbf{a}\\mapsto\\sum_{i\\in I}\\mathbf{a}^{i}\\,\\mathbf{F}_{i},\\quad\\mathbf{a}%\\in\\mathbb{K}^{I}.

In other words, $\\hat{\\mathbf{F}}$ is a linear form on $\\mathbb{K}^{I}$ that takesvalues in $\\mathcal{U}$ instead of $\\mathbb{K}$ . We use row vectors to representlinear forms, and that’s why we write the frame as a row vector.

We call $\\mathbf{F}$ a coordinate frame (equivalently, a basis), if $\\hat{\\mathbf{F}}$ isan isomorphism of vector spaces. Otherwise we call $\\mathbf{F}$ degenerate,incomplete, or both, depending on whether $\\hat{\\mathbf{F}}$ fails to be,respectively, injective and surjective.

Coframes and coordinates.

In cases where $\\mathbf{F}$ is a basis, the inverse isomorphism

\\hat{\\mathbf{F}}^{-1}:\\mathcal{U}\\rightarrow\\mathbb{K}^{I}

is called the coordinatemapping. It is cumbersome to work with this inverse explicitly, andinstead we introduce linear forms $\\mathbf{x}^{i}\\in\\mathcal{U}^{*},\\;i\\in I$ definedby

\\mathbf{x}^{i}:\\mathbf{u}\\mapsto\\hat{\\mathbf{F}}^{-1}(\\mathbf{u})(i),\\quad%\\mathbf{u}\\in\\mathcal{U}.

Each $\\mathbf{x}^{i},\\;i\\in I$ is called the $i^{\\text{th}}$ coordinate functionrelative to $\\mathbf{F}$ , or simply the $i^{\\text{th}}$ coordinate³³Strictly speaking, we should be denote the coframeby $\\mathbf{x}_{\\mathbf{F}}$ and the coordinate functions by $\\mathbf{x}^{i}_{\\mathbf{F}}$ so asto reflect their dependence on the choice of reference frame.Historically, writers have been loath to do this, preferring acouple of different notational tricks to avoid ambiguity. Thecleanest approach is to use different symbols, e.g. $\\mathbf{x}^{i}$ versus $\\mathbf{y}^{j}$ , to distinguish coordinates coming from different frames.Another approach is to use distinct indexing sets; in this way theindices themselves will indicate the choice of frame. Say we havetwo frames $\\mathbf{F}:I\\rightarrow\\mathcal{U}$ and $\\mathbf{G}:J\\rightarrow\\mathcal{U}$ 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 $\\mathbf{x}^{i}$ for coordinates relative to $\\mathbf{F}$ and $\\mathbf{x}^{j}$ forcoordinates relative to $\\mathbf{G}$ . That’s the way it was done in allthe old-time geometry and physics papers, and is still largely theway physicists go about writing coordinates. Be that as it may, thenotation has its problems and is the subject of long-standingcontroversy, named by mathematicians the debauche of indices. Theproblem is that the notation employs the same symbol, namely $\\mathbf{x}$ ,to refer to two different objects, namely a map with domain $I$ andanother map with domain $J$ . In practice, ambiguity isavoided because the old-time notation never refers to thecoframe (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 witha little practice it’s a perfectly serviceable and useful way tocommunicate.. In this way we obtain a mapping

\\mathbf{x}:I\\rightarrow\\mathcal{U}^{*},\\quad i\\mapsto\\mathbf{x}^{i}

called the coordinate coframe or simply a coframe. The forms $\\mathbf{x}^{i},\\,i\\in I$ give a basis of $\\mathcal{U}^{*}$ . It is the dual basis of $\\mathbf{F}_{i},\\,i\\in I$ , i.e.

\\mathbf{x}^{i}(\\mathbf{F}_{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, wewrite the coframe as a column vector whose components are thecoordinate functions:

\\begin{pmatrix}\\mathbf{x}^{1}\\\\\\mathbf{x}^{2}\\\\\\vdots\\\\\\mathbf{x}^{n}\\end{pmatrix}.

We identify of $\\hat{\\mathbf{F}}^{-1}$ and $\\mathbf{x}$ with the above column-vector.This is quite natural because all of these objects are in naturalcorrespondence with a $\\mathbb{K}$ -valued functions of two arguments,

\\mathcal{U}\\times I\\rightarrow\\mathbb{K},

that maps an abstract vector $\\mathbf{u}\\in\\mathcal{U}$ and an index $i\\in I$ to a scalar $\\mathbf{x}^{i}(\\mathbf{u})$ , calledthe $i^{\\text{th}}$ component of $\\mathbf{u}$ relative to the reference frame $\\mathbf{F}$ .

Change of frame.

Given two coordinate frames $\\mathbf{F}:I\\rightarrow\\mathcal{U}$ and $\\mathbf{G}:J\\rightarrow\\mathcal{U}$ , one can easily show that $I$ and $J$ musthave the same cardinality. Letting $\\mathbf{x}^{i},\\,i\\in I$ and $\\mathbf{y}^{j},\\,j\\in J$ denote the coordinates functions relative to $\\mathbf{F}$ and $\\mathbf{G}$ ,respectively, we define the transition matrix from $\\mathbf{F}$ to $\\mathbf{G}$ tobe the matrix

\\mathcal{M}:I\\times J\\rightarrow\\mathbb{K}

with entries

\\mathcal{M}^{j}_{i}=\\mathbf{y}^{j}(\\mathbf{F}_{i}),\\quad i\\in I,\\;j\\in J.

An equivalent description of the transition matrix is given by

\\mathbf{y}^{j}=\\sum_{i\\in I}\\mathcal{M}^{j}_{i}\\,\\mathbf{x}^{i},\\quad\\mbox{for% all }j\\in J.

It is also the custom to regard the elements of $I$ as indexing thecolumns 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}\\mathcal{M}^{\\bar{1}}_{1}&\\ldots&\\mathcal{M}^{\\bar{1}}_{n}\\\\\\vdots&\\ddots&\\vdots\\\\\\mathcal{M}^{\\bar{n}}_{1}&\\ldots&\\mathcal{M}^{\\bar{n}}_{n}\\end{pmatrix}=\\begin%{pmatrix}\\mathbf{y}^{\\bar{1}}\\\\\\vdots\\\\\\mathbf{y}^{\\bar{n}}\\end{pmatrix}\\begin{pmatrix}\\mathbf{F}_{1}&\\ldots&\\mathbf{%F}_{n}\\end{pmatrix}.

In this way we can describe the relation between coordinatesrelative to the two frames in terms of ordinary matrix multiplication.To wit, we can write

\\begin{pmatrix}\\mathbf{y}^{\\bar{1}}\\\\\\vdots\\\\\\mathbf{y}^{\\bar{n}}\\end{pmatrix}=\\begin{pmatrix}\\mathcal{M}^{\\bar{1}}_{1}&%\\ldots&\\mathcal{M}^{\\bar{1}}_{n}\\\\\\vdots&\\ddots&\\vdots\\\\\\mathcal{M}^{\\bar{n}}_{1}&\\ldots&\\mathcal{M}^{\\bar{n}}_{n}\\end{pmatrix}\\begin{%pmatrix}\\mathbf{x}^{1}\\\\\\vdots\\\\\\mathbf{x}^{n}\\end{pmatrix}

Notes.

The term frame is often used to refer to objects thatshould properly be called a moving frame. The latter can bethought of as a , or functions takingvalues in the space of all frames, and are fully described elsewhere.The confusion in terminology is unfortunate but quite common, and isrelated to the questionable practice of using the word scalarwhen referring to a scalar field (a.k.a. scalar-valuedfunctions) and using the word vector when referring to a vector field.

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

Title	frame
Canonical name	Frame
Date of creation	2013-03-22 12:39:42
Last modified on	2013-03-22 12:39:42
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	13
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A03
Related topic	Vector2
Related topic	TensorArray
Related topic	BasicTensor
Defines	coframe
Defines	component
Defines	coordinate
Defines	transition matrix
Defines	polyad