ENTE PER LE NUOVE TECNOLOGIE,L'ENERGIA E L'AMBIENTE
Associazione EURATOM-ENEA sulla Fusione
ABSTRACT TENSOR ALGEBRA AND APPLICATIONS: AN INTRODUCTION
CAMILLO LO SURDO
ENEA – Unità Tecnico-Scientifica FusioneCentro Ricerche Frascati
RT/2002//FUS
I contenuti tecnico-scientifici dei rapporti tecnici dell'ENEA rispecchiano l'opinione degli autori enon necessariamente quella dell'Ente.
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.
Tensor Algebra over a Pitagorean... 11
(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.
12 Tensor Algebra over a Pitagorean...
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
Tensor Algebra over a Pitagorean... 13
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
14 Tensor Algebra over a Pitagorean...
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.
Tensor Algebra over a Pitagorean... 15
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
16 Tensor Algebra over a Pitagorean...
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.
Tensor Algebra over a Pitagorean... 17
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.
18 Tensor Algebra over a Pitagorean...
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
Isotropic Tensors 21
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.
22 Isotropic Tensors
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)) ,
Isotropic Tensors 23
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.
24 Isotropic Tensors
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)
Isotropic Tensors 25
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).