frame (original) (raw)

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 (http://planetmath.org/Vector2) and the more general abstract vectors (http://planetmath.org/VectorSpace).

Frames and bases.

Let 𝒰 be a finite-dimensional vector space over a field 𝕂, and let I be a finite, totally ordered set of indices11It 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., e.g. (1,2,…,n). We will call a mapping 𝐅:I→𝒰 a reference frame, or simply a frame. To put it plainly, 𝐅 is just a list of elements of 𝒰with indices belong to I. We will adopt a notation to reflect this and write 𝐅i instead of 𝐅⁢(i). Subscripts are used when writing the frame elements because it is best to regard a frame as a row-vector22It is customary to use superscripts for thecomponents of a column vector, and subscripts for the components of a row vector. This is fully described in the vector entry (http://planetmath.org/Vector2). whose entries happen to be elements of 𝒰, and write

This is appropriate because every reference frame 𝐅 naturally corresponds to a linear mapping 𝐅^:𝕂I→𝒰 defined by

In other words, 𝐅^ is a linear form on 𝕂I that takes values in 𝒰 instead of 𝕂. We use row vectors to represent linear forms, and that’s why we write the frame as a row vector.

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

Coframes and coordinates.

In cases where 𝐅 is a basis, the inverse isomorphism

is called the coordinate mapping. It is cumbersome to work with this inverse explicitly, and instead we introduce linear forms 𝐱i∈𝒰*,i∈I defined by

Each𝐱i,i∈I is called the ith coordinate functionrelative to 𝐅, or simply the ithcoordinate33Strictly speaking, we should be denote the coframe by 𝐱𝐅 and the coordinate functions by 𝐱𝐅i 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. 𝐱i versus𝐲j, 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 𝐅:I→𝒰 and 𝐆:J→𝒰 withI and J distinct finite sets. We stipulate that the symbol irefers to elements of I and that j refers to elements of J, and write 𝐱i for coordinates relative to 𝐅 and 𝐱j for coordinates relative to 𝐆. 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 𝐱, 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 thef⁢(x) notation. It creates some confusion for beginners, but with a little practice it’s a perfectly serviceable and useful way to communicate.. In this way we obtain a mapping

called the coordinate coframe or simply a coframe. The forms 𝐱i,i∈I give a basis of 𝒰*. It is the dual basis of 𝐅i,i∈I, i.e.

where δji 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:

We identify of 𝐅^-1 and 𝐱 with the above column-vector. This is quite natural because all of these objects are in natural correspondence with a 𝕂-valued functions of two arguments,

that maps an abstract vector𝐮∈𝒰 and an index i∈I to a scalar 𝐱i⁢(𝐮), called the ith component of 𝐮 relative to the reference frame𝐅.

Change of frame.

Given two coordinate frames 𝐅:I→𝒰 and𝐆:J→𝒰, one can easily show that I and J must have the same cardinality. Letting 𝐱i,i∈I and 𝐲j,j∈J denote the coordinates functions relative to 𝐅 and 𝐆, respectively, we define the transition matrix from 𝐅 to 𝐆 to be the matrix

with entries

An equivalent description of the transition matrix is given by

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,…,n) andJ=(1¯,2¯,…,n¯), we can write

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

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 , 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 scalarwhen 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.