Kusse B.R., Westwig E.A. Mathematical Physics: Applied Mathematics for Scientists and Engineers

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INTRODUCTION

TENSORS

AXB=

before,

the

transformation matrix can also

determined from the basis vectors

of the two coordinate systems. For the general curvilinear case, the elements of

[a]

are

a..

(q;

qj).

(4.101)

The above manipulations

are

fast and easy using subscript notation. It might be a

useful exercise to go through the same steps using

just

matrices

convince yourself

the subscript notation is more efficient.

4.6

PSEUDO-OBJECTS

we consider only transformations that involve rigid rotations or translations, there

is no way to change a right-handed system into a left-handed system, or vice-versa.

To change handedness requires a reflection. Transformations that involve reflections

require the introduction

the so-called “pseudo”-objects. Pseudoscalars, pseudo-

vectors, and pseudotensors

are

very similar to their “regular” counterparts, except

for their behavior when reflected.

easy way to demonstrate the distinction is

by closely examining the cross product

two regular vectors in right-handed and

left-handed systems. Another way that emphasizes the reflection properties of the

transformation is the mirror test, which

presented

Appendix E.

4.6.1

Pseudovectors

Consider the right-handed Cartesian coordinate system shown in Figure 4.4. Picture

two regular vectors in

this

system, oriented along the first two basis vectors:

(4.102)

(4.103)

By “regular,” we mean that the components of these vectors obey Equation 4.25 when

we transform the coordinates.

The cross product between

and can be formed using the determinant,

A,B,&,

(4.104)

iet

Figure

4.4

The Right-Handed System

PSEUDO-OBJECTS

AxB

Figure

4.5

Vectors

the Right-Handed System

or, equivalently, using the Levi-Civita symbol:

(4.105)

The resulting vector is shown in Figure 4.5. Notice how the direction of

is given

by the standard right-hand rule.

you point the fingers of your hand in the direction

and then curl them to point along

your thumb will point in the direction of the

cross product. Keep in mind that the cross product is not commutative.

the order

of the operation is reversed, that is

you form

the result points in exactly the

opposite direction.

Now consider the left-handed system shown in Figure 4.6, with coordinates and

basis vectors marked with primes to distinguish them from the coordinates and basis

vectors of the right-handed system. This system results from a simple inversion

the I-axis of the unprimed system. It can

also

looked at as a reflection

the right-

handed system about its x2x3-plane. The equations relating the primed and unprimed

coordinates are

(4.106)

Figure

4.6

The

Left-Handed System

INTRODUCTION TO TENSORS

that the transformation matrix becomes

-1

0 0

[a]=

[

The regular vectors

and

in the primed coordinate system are simply

-A,

6;.

(4.107)

(4.108)

(4.109)

We just wrote these results down because they were obvious. Remember, formally,

they can be obtained by applying

[a]

to the components of the unprimed vectors. The

matrix multiplication gives

(4.1

10)

and

-1

[:I]

[

[lJ

[;,I

It is important to remember that the vectors are the same physical objects in both

coordinate systems. They are just being expressed in terms

different components

and different basis vectors.

Now form the cross product of

and

in the left-handed system. To do this we

will use the same determinant relation:

(4.111)

AXB=

-A,

0 0

-A,

B,C&,

(4.112)

Let

or in terms of the Levi-Civita symbol:

Eijk

€123

-A,

6:.

(4.113)

The vectors

and

and the cross product

are shown in Figure 4.7 for the

left-handed coordinate system. Notice now, the right-hand rule used earlier

longer

works to find the direction

the cross product.

we define the cross product using

the determinant in Equation 4.112, then we must use a left-hand rule if we are

left-handed coordinate system.

There is something peculiar here. Compare Figures 4.7 and 4.5 notice that while

and

point in the same directions in the two systems, their cross product does not!

PSEUDO-OBJECTS

Figure

4.7

Vectors

The

Left

Handed System

By changing the handedness

the coordinate system, we have managed to change

the vector

Let's look at this cross product from another point of view.

the unprimed

components of the quantity

given in Equation 4.104, are transformed into the

primed system using the

[a]

matrix, as one would

for regular vector components,

we obtain

[

[At,,]

[At,]

(4.1 14)

Combining these components with the appropriate basis vectors gives for the cross

product vector

-1

$4.

(4.1

15)

This result disagrees with Equation 4.1 12 by a

minus

sign.

get around this difficulty,

a quantity formed by the cross product of two regular vectors is called

apsedovector.

Pseudovectors are

also

commonly called axial vectors, while regular vectors are

called polar vectors. If

is a regular vector it transforms according to Equation 4.25.

However, if

is a pseudovector, its components transform according to

bidet

Viari-

(4.1

16)

In this way Equation 4.1 14 becomes

B);

-1

(A x

B);

0 0

A,B,

'43"

[(AXE1:]

[

]

[

]

giving

(4.117)

-A,

$;,

(4.1 18)

in agreement with Equations 4.1 12 and 4.1 13.

INTRODUCTION

TENSORS

summarize, if

is a regular vector its components transform as

a,i.

If instead it is a pseudovector, it components transform as

(4.1

19)

(4.120)

If the handedness of two orthonormal coordinate systems is the same, a transformation

between them will have

and vectors and pseudovectors will both transform

normally. If the systems have opposite handedness,

la/&

-1

and vectors will

transform normally but pseudovectors will flip direction.

vector generated by a

cross product of two regular vectors

actually a pseudovector.

It is tempting to think that all this balderdash is somehow a subtle sign error

embedded in the definition of the cross product.

some cases, this is correct. For

example, when we define the direction of the magnetic field vector, which turns out

to be a pseudovector, we have implicitly made an arbitrary choice of handedness

that must be treated consistently. Another example is the angular momentum vector,

which is defined using a cross product. While you could argue that the “pseudoness”

of these two examples is just

problem with their definition, there are cases where

you cannot simply explain

this

property away. It

possible to design situations where

an experiment and its mirror image do not produce results which are simply the mirror

images of each other. In fact, the

Nobel

Prize was won

Lee and Yang for analyzing

these counterintuitive

violations

purity conservation.

The classic experiment was

first performed by

Wu,

who showed

this

effect with the emission of beta particles

from Cobalt-60, under the influence of the weak interaction.

4.6.2

Pseudoscalars

The ideas that led us to the concept

pseudovectors apply to scalars as well.

proper

scalar is invariant to any change of the coordinate system. In contrast, a pseudoscalar

changes sign if the handedness of the coordinate system changes.

pseudoscalar

involved in a transformation, governed by the transformation matrix

[a],

will obey

(4.121)

good example of a pseudoscalar derives from the behavior of the cross product

operation. The volume of a three-dimensional parallelogram, shown in Figure 4.8,

can be written as

Volume

(4.122)

right-handed system, the vector formed by

will point in the upward

direction.

in a right-handed system,

(4.123)

PSEUDO-OBJECTS

In a left-handed system,

points downward, and consequently

(4.124)

Interpreted in this way, the volume of a parallelogram is a pseudoscalar.

4.6.3

Pseudotensors

Pseudotensors are defined just as you would expect. Upon transformation, the com-

ponents of a pseudotensor obey

which

exactly the same as a regular tensor, except for the

laid,,

term.

Again we turn to the cross product to find a good example. Consider two coordinate

systems. One, the unprimed system, is a right-handed system, and the other, with

primed coordinates, is left-handed. Using the Levi-Civita symbol in both coordinate

systems to generate the cross product of

and

gives the relation

The minus sign occurs because, as we showed earlier, the physical direction

the

cross product is different in the two coordinate systems. Now the transformation

properties

regular vectors can be used to find

the

relationship between

Eijk

and

Because

and the basis vectors are all regular vectors, they transform according

to Equation 4.25. Writing the primed components of these vectors in terms

the

unprimed, Equation 4.126 becomes

INTRODUCTION

TENSORS

This

expression is true for arbitrary

and

we obtain the result

Eijk

-ariasjat&.

(4.128)

Keep in mind

this

applies only when the two systems have opposite handedness.

If both systems have the same handedness, the

minus

sign disappears.

Thus

for

the general case of arbitrary transformation between

two

orthonormal systems, the

Levi-Civita symbol components obey

Eijk

blder%%jatk'&t.

(4.129)

Consequently, the Levi-Civita symbol is a pseudotensor.

EXERCISES FOR CHAPTER

Determine the transformation matrix

[a]

that corresponds to a rotation of a two-

dimensional Cartesian system by

30".

Obtain the inverse of

this

matrix

[a]-'

and

evaluate the following operations:

(a)

a;'ajm.

(b)

aij'a,j.

(c)

Uj'Qmj.

Consider the transformation from a standard two-dimensional Cartesian system

(xl,

xz)

to a

primed

system

(xi,

xi)

that results from a reflection about the

-axis,

as shown below:

Express the coordinates

a point

the primed system in terms of the

coordinates of the same point in the unprimed system.

Determine the elements of the transformation matrix

[a]

that takes vector

components from the unprimed system to components in the primed system.

Determine

[a]-'

in three ways:

By inverting the

[a]

matrix found in part

(b).

ii.

By inverting the coordinate equations of part

(a).

iii.

By simply switching the primed and unprimed labels on the coordinates.

EXERCISES

Determine the elements of the transformation matrix

[a]

that generates the Carte-

sian components of a vector from its components in the toroidal system defined

in Exercise

of Chapter

The coordinates of a hyperbolic system

(u,

are related to a set of Cartesian

coordinates

(x,

by the equations

Y‘

2xy

Determine the elements of the transformation matrix

[a]

that takes the Cartesian

components of a vector to the hyperbolic components. Using this transformation

matrix and the position vector expressed in the Cartesian system, express the

position vector in the hyperbolic system.

Consider two curvilinear systems, an unprimed cylindrical system and a primed

spherical system.

(a)

What are the equations relating the cylindrical coordinates to the spherical

coordinates?

(b)

Find the elements of the transformation matrix

[a]

that generates the primed

vector components from unprimed components.

Consider a two-dimensional, primed Cartesian system that is shifted an amount

and rotated an amount

with respect to a two-dimensional unprimed Cartesian

system, as shown in the figure below.

Y’

X’

A-

(a)

Identify the equations that generate the xy-coordinates from the

x’y

’-

(b)

Determine the elements of the transformation mamx

[a]

that generate the

Consider a two-dimensional Cartesian xy-system and a shifted polar p’O’-system,

as shown in the figure below. The origin of the shifted polar system is located at

r,,y

coordinates.

primed components of a vector from the unprimed components.

INTRODUCTION TO TENSORS

(a)

a sketch, pick a point

well

off

the

x-axis

and draw the Cartesian and

shifted polar basis vectors.

(b)

Express the Cartesian coordinates in terms of the shifted polar coordinates.

Invert these equations and express the

shifted

polar coordinates in terms of

the Cartesian coordinates.

(c)

Find the elements of the transformation matrix

[a]

that relates the shifted

polar basis vectors to the Cartesian basis vectors.

(d)

Express the displacement vector

first in the Cartesian system and then in

the

shifted

polar system.

two-dimensional

(u,

elliptical coordinate system can be related to an

(x,

Cartesian system by the coordinate equations

coshucos

sinhusinv.

Determine the elements of the transformation matrix

[a]

that converts vector

components in the Cartesian system to vector components in

this

elliptical sys-

tem. what

[a]-’?

Consider

three

coordinate systems:

(1)

a Cartesian system with coordinates

(XI,

x2, x3)

and basis vectors

(el,

62,

&3);

(2)

a curvilinear system with coordinates

(91,

q2,

q3)

and basis vectors

(81,

q3);

and

(3)

a second curvilinear system

with coordinates

(qi,q;,

4:)

and basis vectors

(qi,

q:,

a:).

Let

the relationships

between the Cartesian coordinates and these curvilinear coordinates be given by

xh3q29q3)

z(q1,

q27

q3)

X’(q;,q:,q:)

Z’M,

s;.

4:).

Y(ql,q2>q3) Y

Y’(4;4;4:)

(4.130)

(a)

Find the general expressions for the elements of the transformation matrix

[a]

that takes vector components

from

one curvilinear system

the other.

(b)

Take one of the curvilinear systems to be the cylindrical system, and the

other to be the elliptical system of Exercise

and specifically determine the

elements of

[a]

and its inverse.

EXERCISES

10.

Consider a two-dimensional Cartesian system with coordinates

XI,

x2)

and basis vectors

(6,

61,C,

62)

and a polar system with coordinates

q1,4

q2)

and basis vectors

(6,

&).

(a)

Express the Cartesian coordinates

(xl,

x2)

in terms of the polar coordinates

(b)

Express the position vector

ii;,

first using the Cartesian basis vectors and then

using the polar basis vectors.

(c)

Determine the elements of the transformation matrix

[a]

that takes vector

components from

the

polar system to vector components in the Cartesian

system. Show that this transformation matrix works for the expressions for

the position vector of part (b) above.

(d)

What are the elements of

[a]-’,

the matrix that generates polar components

from the Cartesian components?

(e)

Determine the elements of the displacement vector

dii;

in the Cartesian and

polar systems.

(f)

second-rank tensor expressed in the Cartesian system takes the following

form

(41,

q2).

What are its elements in the polar system?

11.

Consider the transformation from a two-dimensional Cartesian 12-system to a

Cartesian 1’2’-system, which includes both an inversion and a rotation as shown

in the drawing below.

1’

,/+t

2’

(a)

Express the vector

36’

222

in the 1’2’-system.

(b)

Express

ij@i6j

in the 1 ’2’-system. Do not forget to sum over the repeated

and

subscripts.