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ABSTRACT TENSOR ALGEBRA AND APPLICATIONS: AN INTRODUCTION

CAMILLO LO SURDO

ENEA – Unità Tecnico-Scientifica FusioneCentro Ricerche Frascati

RT/2002//FUS

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The technical and scientific contents of these reports express the opinion of the authors but notnecessarily the opinion of ENEA.

INTRODUZIONE ALL’ALGEBRA TENSORIALE ASTRATTA E ALLE SUE APPLICA-ZIONI

RiassuntoL’Algebra Tensoriale “Astratta” è una branca dell’Algebra Multilineare che, in unionecon l’Analisi Tensoriale in senso stretto, costituisce uno strumento di fondamentaleimportanza per lo studio della maggior parte delle teorie fisico-matematiche, il cosiddettoCalcolo Tensoriale.In quanto segue, forniremo una breve introduzione generale a tale algebra, concentrando-ci poi sulla teoria dei Tensori Isotropi come esempio di sua applicazione di particolareinteresse.

ABSTRACT TENSOR ALGEBRA AND APPLICATIONS: AN INTRODUCTION

Abstract“(Abstract)” Tensor Algebra” is a branch of Multilinear Algebra that, together with“Tensor Analysis” - in its usual “coordinatational” acceptation -, forms the so-called“Tensor Calculus”. This calculus (as a rule to be referred to the pitagorean version ofthe underlying algebra), is of paramount importance for most physico-mathematical theo-ries, from Classical Mechanics of Continuous Material Media to Electromagnetism andGeneral Relativity.In what follows, we shall provide a compact general introduction to the subject, thendwelling upon Isotropic Tensors as a specially important example of application.

Key words: pitagorean linear space; multilinear functional; covariance/contravariance; κ-tensor space; isotropy

Foreword

“(Abstract) Tensor Algebra” is a branch of Multilinear Algebra that,

together with “Tensor Analysis” — in its usual “coordinatational” ac-

ceptation —, forms the so–called “Tensor Calculus”. This calculus (as a

rule to be referred to the pitagorean version of the underlying algebra), is

of paramount importance for most physico–mathematical theories, from

Classical Mechanics of Continuous Material Media to Electromagnetism

and General Relativity.

In what follows, we shall provide a compact general introduction to

the subject, then dwelling upon Isotropic Tensors as a specially important

example of application.

7

1. Tensor Algebra over a Pitagorean, finite–di-

mensional, linear space Xn

Let Xn

be a linear space over R with n ≥ 1 dimensions1 (namely, such as to

contain sets of n, yet not of n+1, linearly independent elements), endowed

with the “Pitagorean” inner product (·). By definition, this product —

an application from X2 ≡ X×X, × ≡ cartesian product, into R — fulfils

the following “Pitagorean Axioms”. With x, y, . . . being generic elements

of X, and α, β . . . generic real numbers, ∀ (x, y, . . .), ∀ (α, β, . . .),

(P1) x · y = y · x;

(P2) (αx) · y = α(x · y);

(P3) x · (y + z) = x · y + x · z;

(P4a) x · x = 0 ⇒ x = OX ; (P4b) x 6= OX ⇒ x · x > 0.

where OX ≡ zero–element of X 2. For shortness, a linear space with

pitagorean inner product will be said “pitagorean” in its turn3.

A number of important properties/theorems hold in a pitagorean

space; e.g., the Schwarz inequality, the triangle inequality, the Schmidt

orthonormalization of a set of (linearly)) independent elements (hence

the existence of an orthormal n–basis in an n–dimensional space), and

the possibility of defining a “norm” of x as (x · x)1/2.

1Since this n will be kept fixed, as a rule it will not be written out in Xn

in what

follows.2The inverse of implication (P4a) follows from (P2). Then x · x ≥ 0 follows from

that inverse and (P4b).3It may be of worth reminding that a linear space over R, with inner product

(·): X2 → R, is “Euclidean” if the product axioms (P1 ÷ P3) are kept unvaried,whereas (P4(a,b)) are replaced by the weaker one

(E4) ∀ y(x · y = 0) ⇒ x = OX .

Thus a pitagorean X is always euclidean, but not the converse. “Pitagorean” is some-times replaced by “properly euclidean”.

8

Tensor Algebra over a Pitagorean... 9

Let now∗

τκ, κ integer ≥ 1, be a real κ–linear form (or κ–form for

shortness) over X; namely, a specific real functional defined over Xκ ≡

X × . . .×X︸︷︷︸

κ times

and linear w.r.t. each of its κ indeterminates — say1x, . . . ,

κx

— when separately taken. This property will be referred to as “κ–linearity”.

For κ = 0, instead,∗

τκ

will be simply defined as a (specific, real) number.

Let us consider the set of the κ–forms over X; within this set, two κ–forms

can be summed (or a κ–form can be multiplied by a real number) in the

usual way, via the values they assume in correpondence to an arbitrary

(ordered) κ–ple {1x, . . . ,

κx}. E.g.,

∗

τκ

+∗

σκ

will be defined as that κ–linear

functional whose value corresponding to the κ–ple {1x, . . . ,

κx} ∈ Xκ is the

sum of the similar values of∗

τκ

and∗

σκ, for every κ–ple of Xκ (κ–linearity

will obviously be preserved in this operation). In what follows, the value

of∗

τκ

corresponding to {1x, . . . ,

κx} will be denoted as

∗

τκ

1x . . .

κx. Plainly, the

set of the κ–forms is a linear space over R, to be denoted as∗

X(κ), in force

of these definitions.

Let {ei}i=1,...,n ≡ {ei} be a (not necessarily orthonormal) basis of X; it

will be convenient to think of it as a fixed, “reference” basis. By definition

of basis, every x ∈ X can be written as xiei, where the xi’s are uniquely

determined real numbers (here and in what follows Einstein’s summa-

tion rule is used). Due to κ–linearity, for any κ–ple {1x, . . . ,

κx} of Xκ,

the corresponding value of∗

τκ,∗

τκ

1x . . .

κx, is equal to

1x i1 . . .

κx iκ ∗

τκei1 . . . eiκ .

The value of∗

τκ

corresponding to {ei1 , . . . , eiκ},∗

τκei1 . . . eiκ , will be called

“covariant component of indices i1, . . . , iκ of the κ–tensor4 τκ

associated

with the κ–form∗

τκ”, and will be denoted as τi1...iκ. There are nκ generally

independent (covariant) components of a κ–tensor whatever.

Now assume that the reference basis be transformed into a new basis

{ei} through the linear nonsingular law

ei = Ljiej ,(1.1)

4For the moment, this is a mere “way of saying”, because the concept of “tensor”has not yet been introduced.

10 Tensor Algebra over a Pitagorean...

where, if L is shorthand for the matrix {Lji}i,j=1,...,n, det L 6= 0; then, the

component set {τi1...iκ} will be transformed into a corresponding compo-

nent set {τ i1...iκ} according to the “κ–ply covariant law”

τ i1...iκ = Lj1i1 . . . Ljκ

iκ τj1...jκ. 5(1.2)

It is not difficult to see that the 2κ–indexed matrix L{κ} ≡ {Lj1i1 . . . Ljκ

iκ}

is nonsingular too, precisely because

det L{κ} = (det L)κnκ−1

.

Due to the assumed pitagoreannes of X, it can be proved that the

quadratic form with symmetric coefficients

gij.= ei · ej(1.3)

in the (real) indeterminates ξi,n∑

1i,jgijξ

iξj, turns out definite–positive;

and so, in particular, writing g for the (symmetric) matrix {gij},

det g > 0 . 6(1.4)

Evidently, the coefficients gij transform according to a 2–ply covariant

law if acted upon by the matrix L{2}, i.e.:

gij = Lhi L

kj ghk .(1.5)

Due to the nonsingularity of g, n elements ej of X exist unique, which

fulfil the n relations

ei = gijej .(1.6)

It is almost immediate to see that these ej’s form a basis in their turn,

the “cobasis” {ej} associated with the basis {ei}7. Still in force of eq.

5In the notations used in eqs (1.1,1.2), one can think of the lower [upper] indices(of the matrices of concern) as “row” [“column”] indices of the standard notation.

6As it is well known, det g > 0 is only one of the n indipendent inequalities whichfollow from the positive–definite character of the quadratic form of above. Also, notethat one would merely find det g 6= 0 in the case of an euclidean X .

7Sometimes, the cobasis associated with the basis {ei} is said “dual” of {ei},although this attribute is not fully appropriate to the present context.


(1.6), one sees that, if {ej} undergoes the covariant transformation (1.1),

{ej} transforms into {ei} according to the “simply contravariant law”:

ei =−1

Lije

j ,(1.7)

where−1

L denotes the inverse of L. (Again, it is immediate to verify that

{ei} is a basis of X). Finally, it is also proved that

ei · ej = δj

i (≡ Kronecker symbol) .(1.8)

In a similar way as it was done for τi1...iκ, the value of∗

τκ

corresponding to

{ei1 , . . . , eiκ},∗

τκei1 . . . eiκ , will be said “contravariant component of indices

i1, . . . , iκ (w.r.t. the reference basis) of the κ–tensor τκ

associated with the

κ–form∗

τκ”, and will be denoted as τ i1...iκ. Due to the κ–linearity of

∗

τκ, in

correspondence with the usual transformation (1.1), the set {τ i1...iκ} will

be turned into a new set {τ i1...iκ} according to the “κ–ply contravariant

law”

τ i1...iκ = Li1j1 . . . Liκ

jκτ j1...jκ ,(1.9)

where the nonsingularity of L{κ} ensures the uniqueness of {τ j1...jκ}.

One can also consider a more general type of component (of τκ),

namely the value of∗

τκ

corresponding to a choice whatever of elements of

{ei} in some positions (...), and of elements of {ej} in the other positions

(...). These components of τκ

are said “mixed”, with covariant [contravari-

ant] indices in the positions (...) [(...)], and are written accordingly. For

instance, considering the 3–tensor τ3, τij

k .=

∗τ3

eiejek is the mixed com-

ponent of τ3

with covariant indices i, j in the 1st, and respectively 2nd,

positions, and contravariant index k in the 3rd position. The transforma-

tion law of a mixed component, w.r.t. the usual transformation (1.1), is

easily deduced. For instance, in the above example, one has

τ ijk = Ll

iLmj

−1

Lknτlm

n .8

8It is essential, in writing a mixed component of a tensor, to keep the index order;thus τij

k cannot simply be written as τkij , a notation that does not exhibit the original

order i, j, k.


Plainly, the inner product between two elements of X is a symmetric

2–form, so the n(n+1)/2 gij’s can be thought of as covariant components

of an associated (symmetric) 2–tensor, say g2

(“first fundamental” 2–

tensor associated with X).

Let us now consider gij as a (symmetric) matrix element, under the

usual convention as for the index position (left ≡ row, right ≡ column).

By definition,∑

jgij

−1gjk = δik, so gij =

∑

h,kgihghk

−1gkj ≡

∑

h,kgih

−1ghkgkj. But

gij = ei · ej =∑

h,kgihgjke

h · ek (eq. (1.6)), hence 0 =∑

h,kgihgjk

(−1ghk − ghk

)

,

where we have written eh · ek as ghk ≡ gkh, the (2–ply) contravariant

components of g. Since the 4–index matrix {gihgjk} is nonsingular too

(its determinant is (det g)2n), we get:

−1ghk = ghk ,(1.10)

so∑

j

gijgjk ≡ gijg

jk = δki .(1.11)

Finally, we define the mixed components of g2

according to gik = gk

i ≡

gki

.= ei · ek; but this is equal to gije

j · ek ≡ gijgjk, and we conclude that

gki = δk

i .(1.12)

The above results allow us to establish the following simple rule. If we

“contract” a component of a given κ–tensor τκ, in which i is a covari-

ant [contravariant] index, and j is not an index, with gij [gij] (i.e. if

we multiply it by gij [gij] and sum over i) that index i becomes con-

travariant [covariant] keeping the same position with “name” j, i.e.:

τ · · ·i · · · gij = τ · · ·j · · · (and similarly), where the dots at intermediate

height represent covariant or contravariant indices, all of which different

from j. Also, we observe that a possible symmetry, or antisymmetry, of

a tensor component w.r.t. a given position pair, persist however those

indices are displaced vertically. Namely, for instance making reference to


a symmetry i � k,

τ · · ·i · · ·k · · · = τ · · ·k · · ·i · · · ⇔ τ · · ·i · · ·k · · · = τ · · ·k · · ·i · · · ,

and so on. Furthermore, the above properties do not depend on the basis

(namely they are intrinsic properties of the tensor considered); indeed,

they persist through a generic (linear, nonsingular) transformation of the

basis.

What remains to be specified is the definition of the κ–tensor τκ

as an

element of a convenient linear space, in terms of the κ–form∗

τκ, and for the

given X. The answer is immediate for κ = 1: we state that τ1

.= τ iei ≡ τie

i,

with τi [τ i] being defined the usual way starting from∗τ1. Thus τ

1∈ X: the

linear space of the 1–tensors, or “vectors”, associated with X, we shall

denote as X (1), simply coincides with X, and the two spaces X (1) ≡ X

and∗

X (1) (of the 1–forms over X) are isomorphic as linear spaces. The

inner product of two vectors (σ1, τ

1), σ

1· τ1, is immediately evaluated and

turns out equal to σiτi ≡ σiτi. This evidently fulfils the inner–product

axioms (P1) ÷ (P4), as it must be. We can also define the inner product

between the corresponding 1–forms∗σ1

and∗τ1

according to∗σ1·∗τ1

.= σiτ

i.

Thus X (1) and X∗ (1) are isomorphic as “linear spaces with inner product”

too, in particular with σ1· τ

1=

∗σ1·∗τ1.

Before going on beyond κ = 1, it is convenient to extend the algebra

on the set of the linear spaces of the κ–forms∗

X (κ), which at present is

limited to the linear operations in each of the∗

X (κ)’s, and to the inner

product in∗

X (1). Let us consider two forms∗σι∈

∗

X (ι) and∗

τκ∈

∗

X (κ), ι ≥ 0,

κ ≥ 0, and the ordered product (in the usual sense) of the∗

σι

value in

{x1, . . . , xι} by the∗

τκ

value in {xι+1, . . . , xι+κ}, ∀ {x1, . . . , xι+κ} ∈ X ι+κ.

This product defines a (ι + κ)–form, we shall denote as∗

σι

∗

τκ∈

∗

X (ι+κ) and

shall call “algebraic product” of∗

σι

by∗

τκ

(in this order). Plainly, if ι ≥ 1,

κ ≥ 1, not every (ι + κ)–form of∗

X (ι+κ) can be thought of as algebraic

product of a ι–form by a κ–form, and not even as algebraic product of a


ι′–form by a κ′–form under ι + κ = ι′ + κ′, ι′ ≥ 1, κ′ ≥ 1. The set of the

(ι+κ)–forms of type∗

σι

∗

τκ

is a linear space we shall denote as∗

X (ι)∗

X (κ); in

force of the definition∗

X (ι)∗

X (κ) ⊂∗

X (ι+κ). As usual writing∗

X (ι) ×∗

X (κ)

for the cartesian product of∗

X (ι) by∗

X (κ), we have thus defined a binary,

generally noncommutative, operation — the “algebraic multiplication”,

we shall write by ordered justaposition — from∗

X (ι)×∗

X (κ) into (yet not

onto in general)∗

X (ι+κ). For this operation, the usual distributive (both to

the left and to the right)9 and associative properties hold. The associative

property, in particular, allows one to write (∗

σι

∗

τκ)∗

ρλ

=∗

σι(∗

τκ

∗

ρλ) as

∗

σι

∗

τκ

∗

ρλ, so

(∗

X(ι)∗

X(κ))∗

X(λ) =∗

X(ι)(∗

X(κ)∗

X(λ)) as∗

X(ι)∗

X(κ)∗

X(λ); and similarly for the

products of more than three factor–elements and factor–spaces.

In order to define X (2) — the space, assumed linear, of the 2–ten-

sors —, we shall introduce a new binary, generally noncommutative op-

eration — the “tensor multiplication”, to be denoted as⊗, and to be writ-

ten “between” — from X2 ≡ X×X into X (2), by definition fulfilling the

usual distributive axioms. Thus for any (ordered) vector pair (σ1, τ1) ∈ X2,

the tensor product σ1⊗ τ

1belongs to X (2) (yet without the set of such

products — or “bivectors” — filling the whole of X (2) in general). More-

over, we shall postulate that the set of the n2 bivectors ei⊗ej = eij ∈ X(2)

(each of which turns out 2–ply covariant w.r.t. the usual linear nonsin-

gular transformation of the basis), {eij}, 1 ≤ ∀ (i, j) ≤ n, be complete

and linearly independent in X (2) (i.e. be a 2–ply covariant basis of X (2)).

This completely determines X (2): every given element τ2

of X (2) is an “in-

variant” linear combination over {eij}, say τ ijeij, where {τ ij} is a 2–ply

contravariant set of n2, uniquely determined, real numbers. A one–to–

one correspondence (“canonical” correspondence) between the elements

of∗

X(2) and those of X (2) can readily be established by identifying τ ij

and τ ij =∗τ2

eiej; this correspondence makes X (2) and∗

X(2) isomorphic as

9I.e. (∗

σι

+∗

λι

)∗

τκ

=∗

σι

∗

τκ

+∗

λι

∗

τκ

, (c∗

σι

)∗

τκ

= c(∗

σι

∗

τκ

), and similarly for the right–distributive

property.


linear spaces. Furthermore,∗

X(1)∗

X(1) (the space of the algebraic products

of 1–forms) and X (1) ⊗X (1) (the space of the bivectors) are isomorphic

w.r.t. the corresponding algebraic, and respectively tensor, multiplica-

tions, according to the self–evident diagram:{

∗σ1,∗τ1

}alg. mult.7−→

∗σ1

∗τ1

l l l{

σ1, τ1

}tens. mult.7−→ σ

1⊗ τ

1

,

where the third vertical 2–arrow represents the canonical correspondence.

We need two more axioms in order to define X (κ≥3). Precisely, we

shall postulate i) that ⊗ be (distributive and) associative (this will al-

low to deal with objects like ei1 ⊗ ei2 ⊗ . . . ⊗ eiκ ≡ ei1...iκ), and ii)

that {ei1,...,iκ} be a basis of X (κ) (evidently, a κ–ply covariant basis).

Again, the generic κ–tensor of X (κ), the invariant τκ, will be uniquely

expressed as τ i1...iκei1...iκ, where {τ i1...iκ} is a (unique) κ–ply contravari-

ant system of nκ real numbers, which will be identified with the corre-

sponding τ i1...iκ ≡∗

τκei1 . . . eiκ . This creates a one–to–one correspondence

between∗

X(κ) and X (κ), which turn out isomorphic as linear spaces.

Moreover, the tensor multiplication of σι

by τκ, σ

ι⊗ τ

κ(an element of

X(ι) ⊗ X (κ) ⊂ X (ι+κ)) gives σi1...iιτ iι+1...iι+κei1...iι+κ, wich is canonically

associated with the (ι + κ)–form∗

σι

∗

τκ, according to an isomorphism dia-

gram completely similar to the previous one. The covariant basis {ei1...iκ}

can be replaced by the contravariant one {ei1...iκ ≡ ei1 ⊗ ei2 ⊗ · · · ⊗ eiκ},

or even by a generic mixed basis. Note that, whereas the distributive and

associative properties of the algebraic multiplication between multilin-

ear forms follow from the definitions, the same properties of the tensor

multiplication must be assumed axiomatically; in this sense, the algebra

on the set {X (κ)} is constructed by analogy with the one on {∗

X(κ)}, in

such a way that the two algebras are isomorphic w.r.t. all the operations

defined up to now (sum and multiplication by a real number, algebraic,

and respectively tensor, multiplication, and inner multiplication between


1–forms, and respectively vectors).

However, the list of the above operations (all of which, but one, are

binary) is not complete. In fact one more binary, generally noncommu-

tative, operation can be introduced between the (ι ≥ 1)–tensor σι

and

the (κ ≥ 1)–tensor τκ, which takes its values in X (ι+κ−2): the “contract-

ing (p, q)–multiplication •

p,q ”, with 1 ≤ p ≤ ι, 1 ≤ q ≤ κ (or rather

the set of the ικ such multiplications). Starting from the corresponding

ι–form∗

σι

and κ–form∗

τκ, this is defined as follows. Let us put ei in the

pth indeterminate position of∗

σι, and ei in the qth indeterminate position

of∗

τκ

(or viceversa) and sum w.r.t. i the resulting products, to obtain

Σi∗

σι. . . e

pi . . .

∗

τκ

. . . eq

i . . . (in the lower row, we have shown the indetermi-

nate positions)10. The above sum is a (ι + κ − 2)–form, say∗

σι

•

(p,q)

∗

τκ, and

the same form independently of the alternative of interchanging ei and

ei with each other. Finally, by definition σι

•

(p,q) τκ

is the (ι + κ− 2)–tensor

canonically associated with the form.

The contracting multiplication between two tensors can also be for-

gotten in the favour of a more general unary operation acting on a single

(κ ≥ 2)–form (or tensor), the “(r, s)–contraction”, where 1 ≤ r < s ≤ κ.

This is defined as∗

τκ7→ Σi

∗

τκ

. . . eri . . . e

s

i . . . (where again we have shown the

indeterminate position in the lower row), and produces a (κ − 2)–form.

The contracting multiplication∗

σι

•

(p,q)

∗

τκ

of before can then be seen as a

(r, s)–contraction with r = p, s = ι + q, acting on the (ι + κ)–form∗

σι

∗

τκ.

Clearly κ(κ−1)/2 different contractions can be made upon∗τ

κ≥2: repeating

contraction until possible, one ends up with a 0–form if the original κ

was even, and with a 1–form if κ was odd.

Contracting multiplications have been already met in the rule to be

used in order to transform a covariant index of a κ–form component into a

controvariant one (or viceversa), where one (at least) of the factors was∗g2.

10Evidently, this reduces to the inner product between the two 1–forms∗

σι,∗

τι

when

ι = κ = 1.


Endowed with the whole of the defined operations,{

∗

X(κ) ∼ X(κ)

}

κ=0,1,...

(where ∼ means “isomorphic to”) is a special algebra, the tensor algebra

“based on X”. First of all, each X (κ) is a linear space over R; then, from

every ordered pair{

X(ι≥1), X(κ≥1)}

one gets X (ι+κ≥2) by means of a ten-

sor multiplication; and finally, from every X (κ≥2) one reduces to X (κ−2)

by means of a contraction.

A final remark is in order before closing this part (1). As a rule, a

physicist (or a mathematical–physicist) learns the rudiments of tensor

algebra in the wider context of the so–called “Tensor Calculus”, along

the classical “coordinatational” approach of Ricci and Levi Civita. Inside

such a “Ricci Calculus”, has a natural place that minimal “algebraic

outfit” which allows the student to multiply a tensor by a real number, to

sum tensors of equal ranks, to multiply two tensors of arbitrary ranks (in

a given order) and to contract a tensor of rank ≥ 2 11. The whole of these

operations are immediately recognized to coincide with the corresponding

ones we have introduced above on a purely axiomatic basis, without

making recourse to a coordinate system. The generalization attained this

way consists in that both the reference basis and its linear nonsingular

transformations (see eq. (1.1)) turn out coordinate–dependent in general.

The situation is easily clarified as follows. Let (y1, . . . , yn) be standard

(real) cartesian coordinates of a point P ≡ Σyhyh of a n–dimensional

geometric space Cn

(yh ≡ hth unit vector).

Let

xi = xi(yh) , i, h = 1, . . . , n ,(1.13)

be a nonsingular transform of (continuity) class C1 about some refer-

ence point x0

� y0, hence det

(∂xi

∂yh

)

|y0

6= 0. Evidently, the n vectors of Cn,

∂P∂xi ≡ ∂iP , are linearly independent at y

0, and can be used as elements

of a basis of an associated n–dimensional linear space Yn

. In front of a

11However, the familiarity with handling the operations of an algebra is quite dif-ferent a thing from understanding that algebra as a formal structure.


generic nonsingular C1 transformation xi� xi about x

0, det

(∂xi

∂xj

)

|x0

6= 0,

one has ∂P∂xi ≡ ∂iP = ∂xj

∂xi ∂jP . This can be interpreted as an 1–covariant

transformation law of the ∂jP ’s (at x0). Furthermore, since Σhdy

2

h =

Σh∂yh

∂xi dxi ∂yh

∂xj dxj ≡ ∂iP ·∂jPdxidxj (where · now denotes the usual scalar

product in Cn) is a positive-definite binary form in the dx’s, its (symmet-

ric) coefficients ∂iP ·∂jP can be interpreted as covariant components of a

(first) fundamental 2–tensor (indeed, det(∂iP ·∂jP ) > 0) associated with

the basis {∂iP}: it will be enough to identify the scalar product in Cn with

the inner product in the (pitagorean) Yn

. Furthermore, it is easily checked

that {∇xj}, j = 1, . . . , n, where ∇.= Σyh

∂∂yh

, is the cobasis of the basis

{∂iP}: in fact, ∂iP = ∂iP · ∂jP∇xj because ∂jP∇xj ≡ ∇P ≡ Σhyhyh ≡

the unit dyadics.

In conclusion, it is easy to verify that all axioms of the local (≡ at x0)

(pitagorean) tensor algebra over Yn

are satisfied with the identifications

Yn≡ X

n(the n–dim linear space introduced at the beginning of this text),

together with the corresponding inner products, ei ≡ ∂iP , Lji = ∂xj

∂xi , etc.

What we have thus established is a local (≡ coordinate dependent) tensor

algebra over Yn

, at P ≡ P0. In particular, if xi = yi, ∂iP = yi, Y

n= C

n,

and the coordinate dependence disappears.

All of the above can be generalized to the case where the inverse of

the C1 transformation (1.13), yh = yh(xi), is replaced by a similar one,

say (1.13bis), with i only running through 1, . . . , m < n. The nonsingu-

larity at x0

then means that the m × n matrix(

∂yh

∂xi

)

has characteristic

m at x0≡ {x

0

1, . . . , x0

m}. The C1 transformation (1.13bis) defines an

m–dimensional manifold “immersed” in Cn, and the nonsingular system

{∂iP}i=1,...,m spans the (local ≡ “at x0”) tangent (m–dim) space Y

mof the

manifold.

Isotropic Tensors 19

2. Isotropic Tensors

Having expounded the foundations of tensor algebra, we shall now il-

lustrate a very important — from both the practical and conceptual

standpoints — application of it: the Isotropic Tensor Theory.

Roughly speaking, an “Isotropic Tensor” is one that is not affected

by a rotation of the underlying space X. Technically, this means that a

(κ ≥ 1)–tensor τκ

is isotropic if its (say, covariant) components τ i1...iκ’s

resulting from the transformation (1.2) when the generic nonsingular

matrix L ≡ {Lji} specializes into a “proper” rotation R = {Rj

i}, are

absolute invariants w.r.t. the group of the associated nonsingular 2κ–

index matrices R{κ} = {Rj1i1 . . . Rjκ

iκ}; namely when

Rj1i1 . . . Rjκ

iκ τj1...jκ≡ τi1...iκ(2.1)

identically ∀R ∈ O+(n) (the group of proper rotations of a n–dim space),

and for every index application i of {1, . . . , n} in itself. The proper rota-

tions, or rotations tout court, or “congruent orthogonal transformations”,

form a subgroup of the more general “orthogonal transformations”, since

they exclude reflections. The following obvious inclusions among groups

are valid (for a given n ≥ 1): GL(n) (linear–nonsingular transforma-

tion group) ⊃ O(n) (orthogonal group) ⊃ O+(n) (congruent–orthogonal

group) ≡ rotation group. Similar group inclusions exist among the cor-

responding “powered” transformations with exponent {κ}12.

It is worth to remind that the orthogonal transformations are those

linear nonsingular transformations which map any orthonormal basis

onto an orthonormal basis. Thus the related matrix, say A.= {Aj

i}, must

fulfil ΣhAhi A

hk = δik; and so, due to a well–known theorem of matrix alge-

bra, (det A)2 = 1. Of course the latter relation is not sufficient, in general,

to ensure the orthogonality of A. On the other hand, a linear nonsingular

12Usually, generic (i.e. not necessarily linear) transformations of a set onto itself, likethose under consideration, are called “substitutions”, and sometimes “permutations”,of that set.

20 Isotropic Tensors

transformation whose matrix has positive determinant is said “congru-

ent”. Thus the matrix of a (proper) rotation, a congruent–orthogonal

transformation by definition, has determinant = +1.

In the language of Invariant Theory, f is an absolute invariant w.r.t.

a (linear) substitution group G over a set D if

∀Γ(∈ G)∀ x(∈ D)[f(Γx) = f(x)] .(2.2)

We can immediately translate the definition (2.1) in this form by iden-

tifying x with {ei} ≡ {ei1 , . . . , eiκ} (where i is any index–application of

{1, . . . , n} in itself), Γ with the 2κ–index matrix R{κ}, or more explicitly

with (Rji )

{κ} (here j will eventually be contracted), G with O+(n), and

f with the κ–form associated with τκ; and finally, D with the basis set,

for every index application i.

By definition, the property of being isotropic, for a τκ

whatever, is

linear (if several κ–tensors are isotropic, every linear combination of them

is isotropic as well); so, for given n ≥ 2, κ ≥ 1, and given X ≡ Xn

, the

n–dim pitagorean linear space over R which supports the linear space

X(κ), the set of all the isotropic κ–tensors is a (proper) linear manifold

I(X, κ) of X (κ) (as usual we shall neglect the explicit transcription of

n). Obviously, this I(X, κ) has dimension < nκ. Due to linearity, to get

this basis it is sufficient to identify a complete set of linearly independent

κ–tensors of I(X, κ).

It is instructive to show that g2

is an isotropic 2–tensor for any X ≡

Xn≥2

. Let {ei} ≡ e be the reference basis; by means of a Schmidt trans-

formation S (which is linear nonsingular), we get an orthonormal basis

e = Se. If this basis undergoes a rotation R, by definition it remains or-

thormal. Acting on it by S−1, we eventually get S−1RSe. Now gij = δij in

e, and this property keeps valid under the action of R. Thus gij becomes

δij under the action of S, keeps equal to δij under R, and finally returns

equal to gij under S−1. On the other hand, e becomes S−1RSe under the

same transformations; but, it can be proved, R and S commute, so the


final basis is Re. In conclusion, the gij’s keep unvaried under a rotation,

q.e.d. This result remains valid for orthogonal transformations in place

of rotations.

A more general (and obvious) statement follows, namely: every 2κ–

tensor whose covariant (say) components are products of the covariant

components of κ tensors g2

is isotropic. It is easily recognized that there

are at most (2κ − 1)!! linearly independent 2κ–tensors of this type: for

instance, for κ = 2, we have (4 − 1)!! = 3 4–tensors, with components

(ikjh) equal to gikgjh, gijgkh and gihgjk.

Every linear combination of such 2κ–tensors is isotropic. If in par-

ticular κ = n, a special linear combination of the related 2n–tensors,

say Ei1...inj1...jn, is the one whose (i1 . . . inj1 . . . jn) covariant component

is defined as

Ei1...inj1...jn

.= det

gi1j1 . . . gi1jn

. . . . . . . . .ginj1 . . . ginjn

(2.3)

Plainly, the value of this isotropic–tensor component does not change

under the interchange i � j, whereas it changes by a factor (−1)p if we

make a permutation of parity p on {i1, . . . , in}, keeping fixed {j1, . . . , jn}

or viceversa (on {j1, . . . , jn} keeping fixed {i1, . . . , in}).

These facts are compatible with the possibility of writing Ei1...inj1...jn

in the form εi1...inεj1...jn, where

εi1...in.= ±εi1 ...in(det g)

12 ,(2.4)

and with εi1...in being in turn the fully antisymmetric symbol13; the sign

in the RHS remaining unspecified for the moment, but fixed for every

choice of the index application i.

However it has not yet proved that the LHS’s of definition (2.4) (once

we have chosen the sign on the right) behave like the (covariant) com-

ponents of an n–tensor. Let L be the matrix which brings the refer-

13The definition of εi1...inpresupposes the choice of a “reference” permutation of

1, 2, . . . , n, for instance 1, 2, . . . , n itself.


ence basis e into e = Le. A theorem on determinants then tells us that

det g = (det L)2 det g (in any case, both det g and det g are > 0). We

shall agree about the following convention, which is compatible with def.

(2.4), and eliminates the alternative due to the sign:

εi1...in = sign det Lεi1...in(det g)1/2 .(2.5)

In particular, for L = identity this gives:

εi1...in = εi1...in(det g)12 ,(2.5bis)

and means that we did actually choose the + sign in def. (2.4).

On specializing eq. (2.5), we get:

{

ε1...n = sign det L(det g)12 and so

ε1...n = (det g)12 , hence

(2.5ter)

ε1...n = sign det L| detL|ε1...n ≡ det Lε1...n ,(2.6)

and finally

εi1...in = det Lεi1...in .(2.7)

Now we show that the εi1...in ’s actually transform according to a covariant

law. We start from the identity

Lj1i1 . . . Ljn

in εj1...jn= εi1...in det L(2.8)

(where the usual Einstein rule of summation has been followed in the

LHS’s) to obtain:

εi1...in = εi1...inε1...n (in force of (2.5, 2.5ter)) =

= εi1...in det Lε1...n = Lj1i1 . . . Ljn

in εj1...jnε1...n (in force of (2.8)) =

= Lj1i1 . . . Ljn

in εj1...jn(in force of (2.5bis, 2.5ter)) ,


which proves our statement. Thus the εi1...in ’s are the covariant compo-

nents of an n–tensor, we shall denote as εn

from now on14.

On the other hand, eq. (2.7) shows that the εi1...in’s are relative in-

variants w.r.t. GL(n), with multiplier det L; hence εn

is an isotropic n–

tensor, because its (covariant) components are absolute invariants w.r.t.

O+(n), the group of the proper rotations (where, as we know, det L = 1).

Moreover, the εi1...inεj1...jn≡ Ei1...inj1...jn

’s are invariant w.r.t. orthogonal

transformations (where | det L| = 1), as it is obvious “a priori” in the

light of definition (2.3), and are covariant components of a 2n–tensor we

shall denote as E2n

. The isotropic nature of this E2n

is evident, because it is

the tensor product of εn

by itself.

One can go further in this sense: a κ–tensor whatever whose (say, co-

variant) components are tensor products of components of g2

(taken p ≥ 0

times) and εn

(taken q ≥ 0 times), in any order, under the requirement

that 2p + nq = κ, is isotropic as well.

One more statement can easily be proved. Let Rji be a rotation matrix.

A theorem of matrix algebra states that det(Rji − δj

i ) = 0 ⇔ Rji = δj

i

∀ (i, j). This is tantamount as saying that, for a rotation Rji different

from the identity, det(Rji − δj

i ) 6= 0, namely that the linear homogeneous

system for the covariant components of any vector τ1, Rj

i τj − τi = 0, has

no nontrivial (eigen)solutions. In other words, no isotropic vector can

exists, for whatever n ≥ 2.

A very important and unexpected fact (not to be proved here) is that

the set of all the κ–tensors obtained as (possibly repeated) products of

g2

and εn, as described above, is complete in the linear space of all the

isotropic κ–tensors; i.e., that every isotropic κ–tensor can be expressed

as a linear combination of those product–tensors. Of course, the set of the

14To define εn, we have followed eq. (2.4) with sign (+). This means that the com-

ponents of εn, in the reference basis, with indices i1, . . . , in in even permutation w.r.t.

1, 2, . . . , n, are > 0. Should we have chosen the reverse, i.e. eq. (2.4) with sign (−), wewould have got the n–tensor opposite to εn. Of course only one of these alternativesis of interest, and we shall always discard the second one.


linearly independent product–κ–tensors of above is finite, hence a basis

for the space of the isotropic κ–tensors can be obtained by just listing

the linearly independent product–κ–tensors of type (g2)p(ε

n)q, 2p + nq =

κ, as they are generated by a convenient “ordering” algorithm; until it

becomes evident, or it is proved, that no new objects of the same type

can be produced that way. In practice, the job is not too complicated for

sufficiently low values of κ and n. The simplest result one gets is that,

for n = κ = 2, the isotropic basis has just two elements, g2

and ε2

(the

linear independence of these tensors being immediately ascertained).

To make some more examples, let κ be > 2 and odd, say κ = κo. Then

both n and q must be odd, say no and qo respectively, and noqo = κo−2p.

If κ is even, instead (say κ = κe), two cases have to be distinguished,

namely: i) even n (≡ ne), with q being either even or odd, but such as

to fulfil neq = κe − 2p, and ii) odd n (≡ no) with q being even (≡ qe),

under noqe = κe− 2p. For example, for κ = 3, noqo = 3− 2p, which gives

no = 3, p = 0, qo = 1, and nothing else; in other words, the isotropic

3–basis is empty for n 6= 3, otherwise it consists of ε3

only.

We conclude this Part 2) by giving the (fully or partially) contracted

products of two εn’s in mixed form. Taking into account that εj1...jn =

εj1...jn(det g)−12 (where εj1...jn ≡ εj1...jn

) one finds:

εk1...knεk1...kn = n! ,(2.90)

εk1...kn−1inεk1...kn−1jn = (n− 1)!δjn

in ,(2.91)

. . . . . . . . . ,

εk1k2i3...inεk1k2j3...jn = 2!δj3...jn

i3...in ,(2.9n−2)

εki2...inεkj2...jn = 1!δj2...jn

i2...in ,(2.9n−1)

εi1...inεj1...jn = δj1...jn

i1...in .(2.9n)


Here the δ’s in the RHS’s are the so–called “Generalized Kronecker

Symbols”, equal to (−1)p if the lower indices are all different from each

other and the upper indices form a permutation of parity p of the lower

ones, and equal to zero in all the other cases. Ignoring the fist row (eq.

(2.90)), which has been added for the completeness’s sake, they are the

mixed components of an isotropic 2–tensor (actually (n−1)! g2, eq. (2.91)),

. . . of a 2(n−2)–tensor (eq. (2.9n−2)), of a 2(n−1)–tensor (eq. (2.9n−1)),

and of an isotropic 2n–tensor (actually E2n

, eq. (2.9n)).

26 General References

General References (in alphabetical order)

R.M. Bower, C.–C. Wang: Introduction to vectors and tensors, 2

vols., Plenum Press, New York–London (1976).

B. Finzi, M. Pastori: Calcolo Tensoriale ed Applicazioni, Zanichelli,

Bologna (1949).

W.H. Greub: Linear Algebra (2nd Ed., 1963), Multilinear Algebra (1967),

Springer, Berlin–New York (1963, 1967).

G.B. Gurevich: Foundations of the Theory of Algebraic Invariants, P.

Nordhoff Ltd, Groningen (1964).

T. Levi Civita: The Absolute Differential Calculus, Blackie & Son,

London–Glasgow (1927).

A. Lichnerowitz: Elements de Calcul Tensorial, Librairie A. Colin,

Paris (1950).

J.A. Schouten: Ricci Calculus. An Introduction to Tensor Analysis

and its Geometrical Applications, Springer, Berlin (1954).

G. Temple: Cartesian Tensors, Methuen & Co. Ltd, London (1960).

B.L. van der Waerden: Algebra, Erster Teil (7. Auflage, 1966), Zweiter

Teil (5. Auflage, 1967), Springer, Berlin (1966, 1967).

H. Weyl: The Classical Groups, their Invariants and Representations,

Princeton Univ. Press, Princeton N.J. (1938).

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