frame

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

(

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 indices

1

1

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.

, 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

𝒰

with indices belong to

I

. We will adopt a notation to

reflect

this
and write

𝐅

i

instead of

𝐅

⁢

(

i

)

. Subscripts are used when
writing the frame

elements

because it is best to regard a frame as a
row-vector

2

2

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 (

http://planetmath.org/Vector2

).

whose entries happen to be elements of

𝒰

,
and write

𝐅

=

(

𝐅

1

,

…

,

𝐅

n

)

.

This is appropriate because
every reference frame

𝐅

naturally corresponds to a

linear
mapping

𝐅

^

:

𝕂

I

→

𝒰

defined by

𝐚

↦

∑

i

∈

I

𝐚

i

⁢

𝐅

i

,

𝐚

∈

𝕂

I

.

In other words,

𝐅

^

is a

linear form

on

𝕂

I

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

𝐅

^

-

1

:

𝒰

→

𝕂

I

is called the coordinate
mapping. It is cumbersome to work with this inverse explicitly, and
instead we introduce linear forms

𝐱

i

∈

𝒰

*

,

i

∈

I

defined
by

𝐱

i

:

𝐮

↦

𝐅

^

-

1

⁢

(

𝐮

)

⁢

(

i

)

,

𝐮

∈

𝒰

.

Each

𝐱

i

,

i

∈

I

is called the

i

th

coordinate

function

relative to

𝐅

, or simply the

i

th

coordinate

3

3

Strictly speaking, we should be denote the coframe
by

𝐱

𝐅

and the coordinate functions by

𝐱

𝐅

i

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.

𝐱

i

versus

𝐲

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

𝐅

:

I

→

𝒰

and

𝐆

:

J

→

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

𝐱

i

for coordinates relative to

𝐅

and

𝐱

j

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 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.

. In this way we obtain a mapping

𝐱

:

I

→

𝒰

*

,

i

↦

𝐱

i

called the

coordinate coframe

or simply a

coframe

. The forms

𝐱

i

,

i

∈

I

give a basis of

𝒰

*

. It is the dual basis of

𝐅

i

,

i

∈

I

, i.e.

𝐱

i

⁢

(

𝐅

j

)

=

δ

j

i

,

i

,

j

∈

I

,

where

δ

j

i

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:

(

𝐱

1

𝐱

2

⋮

𝐱

n

)

.

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

,

𝒰

×

I

→

𝕂

,

that maps an abstract vector

𝐮

∈

𝒰

and an index

i

∈

I

to a scalar

𝐱

i

⁢

(

𝐮

)

, called
the

i

th

component of

𝐮

relative to the reference frame

𝐅

.

Change of frame.

Given two coordinate frames

𝐅

:

I

→

𝒰

and

𝐆

:

J

→

𝒰

, one can easily show that

I

and

J

must
have the same

cardinality

. Letting

𝐱

i

,

i

∈

I

and

𝐲

j

,

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

ℳ

:

I

×

J

→

𝕂

with entries

ℳ

i

j

=

𝐲

j

⁢

(

𝐅

i

)

,

i

∈

I

,

j

∈

J

.

An

equivalent

description of the transition matrix is given by

𝐲

j

=

∑

i

∈

I

ℳ

i

j

⁢

𝐱

i

,

for all

⁢

j

∈

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

,

…

,

n

)

and

J

=

(

1

¯

,

2

¯

,

…

,

n

¯

)

, we can write

(

ℳ

1

1

¯

…

ℳ

n

1

¯

⋮

⋱

⋮

ℳ

1

n

¯

…

ℳ

n

n

¯

)

=

(

𝐲

1

¯

⋮

𝐲

n

¯

)

⁢

(

𝐅

1

…

𝐅

n

)

.

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

(

𝐲

1

¯

⋮

𝐲

n

¯

)

=

(

ℳ

1

1

¯

…

ℳ

n

1

¯

⋮

⋱

⋮

ℳ

1

n

¯

…

ℳ

n

n

¯

)

⁢

(

𝐱

1

⋮

𝐱

n

)

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

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.

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

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