§ 1. Classical Differential Calculus
137
The function
(x, y, u) −→ S(x, u)T (y, x, u)
is obviously not a tensor: the function x
2 is not linear on R or any other field.
Determining the way in which a tensor depends on the coordinates of its
variables with respect to a given basis (a i ) is easy. For example, if T (h, k, u)
is a tensor of type (2, 1), then leaving out the summation signs with respect
to i, j and p – this is Einstein’s summation convention, which I will mostly
use –,
T (h, k, u) = T
h
i a i , k, u
= h
i T (a i , k, u) = h
i T
a i , k
j a j , u
=
= h
i k
j T (a i , a j , u) = h
i k
j T (a i , a j , u p a
p ) =
= h
i k
j u p T (a i , a j , a
p ) ,
and so
T (h, k, u) = T
p
ij h
i k
j u p ,
(1.6)
where the
T
p
ij = T (a i , a j , a
p )
play the role of coefficients, components or coordinates – the terminology
matters little – of T with respect to the basis considered. Conversely, any
function given by a formula of type (6) is clearly trilinear in h, k, u.
Relation (6) easily gives the transformation undergone by the coefficients
T
k
ij under a change of basis from (a i ) to (b p ) given by
b p = ρ
i
p a i ,
(1.7)
where (ρ
i
p ) is the transition matrix from the first basis to the second one. It
suffices to note that for the corresponding dual bases,
b
q = θ
q
j a
j
with ρ
j
p θ
q
j = δ
q
p .
(1.8)
The well-known “ Kronecker delta ” equals 1 or 0 according to whether p and
q are equal or not; indeed, by definition of a dual basis
δ
q
p = b
q (b p ) = θ
q
j a
j
ρ
i
p a i
= θ
q
j ρ
i
p δ
j
i = ρ
j
p θ
q
j .
Formula (6) then shows that
T (b p , b q , b
r ) = ρ
i
p ρ
j
q θ
r
k T
a i , a j , a
k
,
(1.9)
where the summation is over i, j and k. Conversely, it is easy to see that if
we associated numbers T
k
ij transformed according to (9) under basis change
137
The function
(x, y, u) −→ S(x, u)T (y, x, u)
is obviously not a tensor: the function x
2 is not linear on R or any other field.
Determining the way in which a tensor depends on the coordinates of its
variables with respect to a given basis (a i ) is easy. For example, if T (h, k, u)
is a tensor of type (2, 1), then leaving out the summation signs with respect
to i, j and p – this is Einstein’s summation convention, which I will mostly
use –,
T (h, k, u) = T
h
i a i , k, u
= h
i T (a i , k, u) = h
i T
a i , k
j a j , u
=
= h
i k
j T (a i , a j , u) = h
i k
j T (a i , a j , u p a
p ) =
= h
i k
j u p T (a i , a j , a
p ) ,
and so
T (h, k, u) = T
p
ij h
i k
j u p ,
(1.6)
where the
T
p
ij = T (a i , a j , a
p )
play the role of coefficients, components or coordinates – the terminology
matters little – of T with respect to the basis considered. Conversely, any
function given by a formula of type (6) is clearly trilinear in h, k, u.
Relation (6) easily gives the transformation undergone by the coefficients
T
k
ij under a change of basis from (a i ) to (b p ) given by
b p = ρ
i
p a i ,
(1.7)
where (ρ
i
p ) is the transition matrix from the first basis to the second one. It
suffices to note that for the corresponding dual bases,
b
q = θ
q
j a
j
with ρ
j
p θ
q
j = δ
q
p .
(1.8)
The well-known “ Kronecker delta ” equals 1 or 0 according to whether p and
q are equal or not; indeed, by definition of a dual basis
δ
q
p = b
q (b p ) = θ
q
j a
j
ρ
i
p a i
= θ
q
j ρ
i
p δ
j
i = ρ
j
p θ
q
j .
Formula (6) then shows that
T (b p , b q , b
r ) = ρ
i
p ρ
j
q θ
r
k T
a i , a j , a
k
,
(1.9)
where the summation is over i, j and k. Conversely, it is easy to see that if
we associated numbers T
k
ij transformed according to (9) under basis change