Cossey L. Introduction to Mathematical Physics

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RX = X

ijk ijk

With other words ``X is transformed like a

Example:

: piezoelectricity.

As seen at section secpiezo, there exists for piezoelectric materials a relation between the

deformation tensor e

and the electric field h

e = a h

ij ijk k

ijk

is called the piezoelectric tensor. Let us show how previous considerations allow to

obtain following result:

Theorem:

If a crystal has a centre of symmetry, then it can not be piezoelectric.

Proof:

Let us consider the operation R symmetry with respect to the centre, then

R(x x x ) = ( − 1) x x x .

i j k

The symmetry implies

Ra = a

ijk ijk

so that

a = − a

ijk ijk

which prooves the theorem.

Vectorial spaces

Definition

Let K be R of C. An ensemble E is a vectorial space if it has an algebric structure defined

by to laws + and ., such that every linear combination of two elements of E is inside E.

More precisely:

Definition:

An ensemble E is a vectorial space if it has an algebric structure defined by to laws, a

composition law noted + and an action law noted ., those laws verifying:

(E, + ) is a commutative group.

where 1 is the

unity of . law.

Functional space

Definition:

A functional space is a set of functions that have a vectorial space structure.

The set of the function continuous on an interval is a functional space. The set of the

positive functions is not a fucntional space.

Definition:

A functional T of is a mapping from into C.

< T | φ > designs the number associated to function φ by functional

Definition:

A functional T is linear if for any functions φ

and φ

of and any complex numbers λ

and λ

< T | λ φ + λ φ > = λ < T | φ > + λ < T | φ >

1 1 2 2 1 1 2 2

Definition:

Space is the vectorial space of functions indefinitely derivable with a bounded support.

Dual of a vectorial space

Definition

Definition:

Let E be a vectorial space on a commutative field K. The vectorial space of the

linear forms on E is called the dual of E and is noted E

When E has a finite dimension, then E

has also a finite dimension and its dimension is

gela to the dimension of E. If E has an inifinite dimension,

has also an inifinite

dimension but the two spaces are not isomorphic.

===Tensors===

In this appendix, we introduce the fundamental notion of tensor{tensor} in physics. More

information can be found in for instance. Let E be a finite dimension vectorial space. Let

be a basis of E. A vector X of E can be referenced by its components x

is the basis e

X = x e

In this chapter the repeated index convention (or {\bf Einstein summing

convention})

will be used. It consists in considering that a product of two quantities with the same

index correspond to a sum over this index. For instance:

∑

ijk

ikm

∑∑

ijk

ikm

i k

To the vectorial space E correspond a space E

called the dual of E. A element of E

is a

linear form on E: it is a linear mapping p that maps any vector Y of E to a real. p is

defined by a set of number x

because the most general form of a linear form on E is:

p(Y) = xy

A basis e

of E

can be defined by the following linear form

where is one if i = j and zero if not. Thus to each vector X of E of components x

can be

associated a dual vector in E

of components x

X = x e

The quantity

| X | = xx

is an invariant. It is independent on the basis chosen. On another hand, the expression of

the components of vector X depend on the basis chosen. If defines a transformation

that maps basis e

to another basis e'

we have the following relation between components x

of X in e

and x'

of X in e'

This comes from the identification of

and

X = x e

Equations eqcov and eqcontra define two types of variables: covariant variables that are

transformed like the vector basis. x

are such variables. Contravariant variables that are

transformed like the components of a vector on this basis. Using a physicist vocabulary x

is called a covariant vector and x

a contravariant vector.

Let x

and y

two vectors of two vectorial spaces E

and E

. The tensorial product space

is the vectorial space such that there exist a unique isomorphism between the

space of the bilinear forms of

and the linear forms of . A bilinear

form of

is:

b(x,y) = b x x

i j

It can be considered as a linear form of using application from to

that is linear and distributive with respect to + . If e

is a basis of E

and f

basis of E

, then

is a basis of . Thus tensor xy = T

i j ij

is an element of . A

second order covariant tensor is thus an element of . In a change of basis, its

components aij are transformed according the following relation:

Now we can define a tensor on any rank of any variance. For instance a tensor of third

order two times covariant and one time contravariant is an element a of

and noted .

A second order tensor is called symmetric if a = a

ij ji

. It is called antisymmetric is a = −

Pseudo tensors are transformed slightly differently from ordinary tensors. For instance a

second order covariant pseudo tensor is transformed according to:

where det(ω) is the determinant of transformation ω.

Let us introduce two particular tensors.

• The Kronecker symbol δ

is defined by:

It is the only second order tensor invariant in R

by rotations.

• The signature of permutations tensor e

ijk

is defined by:

It is the only pseudo tensor of rank 3 invariant by rotations in R

. It verifies the equality:

e e = δ δ − δ δ

ijk imn jm kn jn km

Let us introduce two tensor operations: scalar product, vectorial product.

• Scalar product a.b is the contraction of vectors a and b :

a.b = a b

i i

• vectorial product of two vectors a and b is:

From those definitions, following formulas can be showed:

Here is useful formula:

==Green's theorem==

Green's theorem allows to transform a volume calculation integral into a surface

calculation integral.

Theorem:

Let ω be a bounded domain of R

with a regular boundary. Let

\partial

\omega</math> (oriented towards the exterior of ω). Let

be a tensor, continuously

derivable in ω, then:{Green's theorem}

Here are some important Green's formulas obtained by applying Green's theorem:

Topological spaces

Definition

For us a topological space is a space where on has given a sense to:

Indeed, the most general notion of limit is expressed in topological spaces:

Definition:

A sequence u

of points of a topological space has a limit l if each neighbourhood of l

contains the terms of the sequence after a certian rank.

Distances and metrics

Definition:

A distance on an ensemble E is an map d from into R

that verfies for all x,y,z in

d(x,y) = 0 if and only if x = y. d(x,y) = d(y,x).

Definition:

A metrical space is a couple (A,d) of an ensemble A and a distance d on A.

To each metrical space can be associated a topological space. In this text, all the

topological spaces considered are metrical space. In a metrical space, a converging

sequence admits only one limit (the toplogy is separated ).

Cauchy sequences have been introduced in mathematics when is has been necessary to

evaluate by successive approximations numbers like π that aren't solution of any equation

with inmteger coeficient and more generally, when one asked if a sequence of numbers

that are ``getting closer do converge.

Definition:

Let (A,d) a metrical space. A sequence x

of elements of A is said a Cauchy sequence if

Any convergent sequence is a Cauchy sequence. The reverse is false in general. Indeed,

there exist spaces for wich there exist Cauchy sequences that don't converge.

Definition:

A metrical space (A,d) is said complete if any Cauchy sequence of elements of A

converges in A.

The space R is complete. The space Q of the rational number is not complete. Indeed the

sequence is a Cauchy sequence but doesn't converge in Q. It converges in

R to e, that shows that e is irrational.

Definition:

A normed vectorial space is a vectorial space equiped with a norm.

The norm induced a distance, so a normed vectorial space is a topological space (on can

speak about limits of sequences).

Definition:

A separated prehilbertian space is a vectorial space E that has a scalar product.

It is thus a metrical space by using the distance induced by the norm associated to the

scalar product.

Definition:

A Hilbert space is a complete separated prehilbertian space.

The space of summable squared functions L

is a Hilbert space.

Tensors and metrics

If the space E has a metrics g

then variance can be changed easily. A metrics allows to

measure distances between points in the space. The elementary squared distance between

two points x

and x + dx

i i

is:

ds = g dx dx = g dx dx

i j ij

i j

Covariant components x

can be expressed with respect to contravariant components:

x = g x

i ij

The invariant xy

can be written

x y = x g y

and tensor like can be written:

Limits in the distribution's sense

Definition:

Let T

be a family of distributions depending on a real parameter α. Distribution T

tends

to distribution T when α tends to λ if:

In particular, it can be shown that distributions associated to functions f

verifying:

converge to the Dirac distribution.

Dual of a topological space