138
IX – Multivariate Differential and Integral Calculus
to each basis (a i ) for E, then the trilinear functional defined by (6) is independent of the basis (a i ).
The convention of placing some indices in an inferior position and others
in a superior one can then be explained as follows. Given a basis (a i ), choose
two linear functionals f = f i a
i and g = g i a
i , a vector h = h
i a i and consider
their tensor product f ⊗ g ⊗ h, i.e. the tensor
S(x, y, u) = f (x)g(y)u(h) (x, y ∈ E , u ∈ E
∗ )
of type (2, 1) like T . Its coefficients in the given base are the numbers
S
k
ij = f (a i ) g (a j ) a
k (h) = f i g j h
k .
Formula (9), therefore, expresses that the coefficients of T transform like
those of f ⊗ g ⊗ h ; besides, by (6), a tensor of type (2, 1) is clearly a linear
combination of such products. Hence, the position of the indices immediately
leads to formula (9).
There are more complicated formulas, but in all cases within the ambit of
tensor calculus, both sides are sums of monomials with indices, for example
A
ij
kh = B
pqi C
j
pq D kh + M pkh N
ijp .
(
∗ )
A formula of this type is a relation between the tensors A, B, C, D, M and
N with coordinates or components in each basis, written A
ij
kh , etc.; it has
relevance only if it holds for any basis. For this and for the formula to have
a chance of being correct, it must satisfy the following conditions:
(a) An index appearing only once in a monomial is a free variable on which
the monomial considered depends; it must occur once and only once
in the inferior or superior position in all the monomials of the relation
considered in order to ensure that with respect to this index, all the
monomials and hence their sum are transformed likewise ;
(b) unless otherwise indicated, an index occurring twice in a monomial is a
summation index, hence a bound or ghost variable on which the monomial does not depend; it must occur once in an inferior position and
once in a superior position in order to ensure that, under basis change,
linear transformations undergone by the monomials relative to these two
indices cancel out ;
(c) a summation index cannot occur more than twice in a given monomial
and cannot occur as a free variable in any other monomials since then,
by rule (a), it would also be occurring as a free variable in all other
monomials.
Exercise. Using formulas such as (9), show that the A
ij
kh given in each
basis by (
∗ ) are indeed the components of a tensor.
Exercise. Let T be a tensor of type (5, 3). Show that the numbers
T
jqr
ijkpq = S
r
ikp
IX – Multivariate Differential and Integral Calculus
to each basis (a i ) for E, then the trilinear functional defined by (6) is independent of the basis (a i ).
The convention of placing some indices in an inferior position and others
in a superior one can then be explained as follows. Given a basis (a i ), choose
two linear functionals f = f i a
i and g = g i a
i , a vector h = h
i a i and consider
their tensor product f ⊗ g ⊗ h, i.e. the tensor
S(x, y, u) = f (x)g(y)u(h) (x, y ∈ E , u ∈ E
∗ )
of type (2, 1) like T . Its coefficients in the given base are the numbers
S
k
ij = f (a i ) g (a j ) a
k (h) = f i g j h
k .
Formula (9), therefore, expresses that the coefficients of T transform like
those of f ⊗ g ⊗ h ; besides, by (6), a tensor of type (2, 1) is clearly a linear
combination of such products. Hence, the position of the indices immediately
leads to formula (9).
There are more complicated formulas, but in all cases within the ambit of
tensor calculus, both sides are sums of monomials with indices, for example
A
ij
kh = B
pqi C
j
pq D kh + M pkh N
ijp .
(
∗ )
A formula of this type is a relation between the tensors A, B, C, D, M and
N with coordinates or components in each basis, written A
ij
kh , etc.; it has
relevance only if it holds for any basis. For this and for the formula to have
a chance of being correct, it must satisfy the following conditions:
(a) An index appearing only once in a monomial is a free variable on which
the monomial considered depends; it must occur once and only once
in the inferior or superior position in all the monomials of the relation
considered in order to ensure that with respect to this index, all the
monomials and hence their sum are transformed likewise ;
(b) unless otherwise indicated, an index occurring twice in a monomial is a
summation index, hence a bound or ghost variable on which the monomial does not depend; it must occur once in an inferior position and
once in a superior position in order to ensure that, under basis change,
linear transformations undergone by the monomials relative to these two
indices cancel out ;
(c) a summation index cannot occur more than twice in a given monomial
and cannot occur as a free variable in any other monomials since then,
by rule (a), it would also be occurring as a free variable in all other
monomials.
Exercise. Using formulas such as (9), show that the A
ij
kh given in each
basis by (
∗ ) are indeed the components of a tensor.
Exercise. Let T be a tensor of type (5, 3). Show that the numbers
T
jqr
ijkpq = S
r
ikp