136
IX – Multivariate Differential and Integral Calculus
Specialists of classical tensor calculus had adopted a system of notation,
now mostly obsolete (including in my Cours d’alg` ebre), that, nevertheless,
had some advantages; it was based on inferior and superior indices and on
the summation convention attributed to Einstein.
In a space E with a basis denoted by (a i ), the first convention consists,
at the simplest level, in writing vectors and linear functionals as follows:
h =
h
i a i , u(h) =
u i h
i
(1.4)
where the h
i are the components of h and the u i = u(a i ) the coefficients of u
with respect to the basis considered. By (2), the second relation (4) can also
be written u(h) =
u i a
i (h), i.e.
u =
u i a
i ,
(1.5)
so that the u i are the coordinates of u ∈ E
∗ with respect to the dual basis
(a
i ) of (a i ) given by a
i (h) = h
i . In calculations involving both vectors and
linear functionals, this notation, with its inferior and superior indices, makes
it possible to immediately detect the nature of the objects discussed;
3 this is
its first advantage.
We generalize this notation to more complex objects, namely tensors. In a
finite-dimensional vector space E, a tensor is a function T of several variables,
some with values in E, the others in E
∗ , and satisfying the same multilinearity
property as a determinant: if all variables except one are held fixed, we get a
linear function of the remaining variable. This property generalizes constantly
used calculation rules in elementary algebra:
(x + y)z = xz + yz , (tx)z = t(xz) .
Calculation rules for tensors are, therefore, the same as those for products,
excepting commutativity.
The function T is generally real or complex valued. T is said to be of
type (p, q), or p times covariant and q times contravariant, if it depends on
p variables in E and q variables in E
∗ . A scalar is a tensor of type (0, 0).
A linear functional is a tensor of type (1, 0). A vector h ∈ E identified with
a linear functional u → u(h) on E
∗ , becomes a tensor of type (0, 1). An
euclidean scalar product (h|k)is a tensor of type (2, 0). A linear map T :
E −→ E becomes a tensor of type (1, 1) if the function T (h, u) = u[T (h)] is
associated to it, and conversely. Given a tensor S(x, u) of type (1, 1) and a
tensor T (x, y, u) of type (2, 1), the function
(x, y, z, u, v) −→ S(x, u)T (y, z, v)
is a tensor of type (3, 2), the tensor product S ⊗ T of S and T . This can be
generalized in an obvious way to other types, provided the variables involved
in both tensors are fully separated.
3 Purists will reply that coordinates can be dispensed with. It does not seem to
be the opinion of physicists.
IX – Multivariate Differential and Integral Calculus
Specialists of classical tensor calculus had adopted a system of notation,
now mostly obsolete (including in my Cours d’alg` ebre), that, nevertheless,
had some advantages; it was based on inferior and superior indices and on
the summation convention attributed to Einstein.
In a space E with a basis denoted by (a i ), the first convention consists,
at the simplest level, in writing vectors and linear functionals as follows:
h =
h
i a i , u(h) =
u i h
i
(1.4)
where the h
i are the components of h and the u i = u(a i ) the coefficients of u
with respect to the basis considered. By (2), the second relation (4) can also
be written u(h) =
u i a
i (h), i.e.
u =
u i a
i ,
(1.5)
so that the u i are the coordinates of u ∈ E
∗ with respect to the dual basis
(a
i ) of (a i ) given by a
i (h) = h
i . In calculations involving both vectors and
linear functionals, this notation, with its inferior and superior indices, makes
it possible to immediately detect the nature of the objects discussed;
3 this is
its first advantage.
We generalize this notation to more complex objects, namely tensors. In a
finite-dimensional vector space E, a tensor is a function T of several variables,
some with values in E, the others in E
∗ , and satisfying the same multilinearity
property as a determinant: if all variables except one are held fixed, we get a
linear function of the remaining variable. This property generalizes constantly
used calculation rules in elementary algebra:
(x + y)z = xz + yz , (tx)z = t(xz) .
Calculation rules for tensors are, therefore, the same as those for products,
excepting commutativity.
The function T is generally real or complex valued. T is said to be of
type (p, q), or p times covariant and q times contravariant, if it depends on
p variables in E and q variables in E
∗ . A scalar is a tensor of type (0, 0).
A linear functional is a tensor of type (1, 0). A vector h ∈ E identified with
a linear functional u → u(h) on E
∗ , becomes a tensor of type (0, 1). An
euclidean scalar product (h|k)is a tensor of type (2, 0). A linear map T :
E −→ E becomes a tensor of type (1, 1) if the function T (h, u) = u[T (h)] is
associated to it, and conversely. Given a tensor S(x, u) of type (1, 1) and a
tensor T (x, y, u) of type (2, 1), the function
(x, y, z, u, v) −→ S(x, u)T (y, z, v)
is a tensor of type (3, 2), the tensor product S ⊗ T of S and T . This can be
generalized in an obvious way to other types, provided the variables involved
in both tensors are fully separated.
3 Purists will reply that coordinates can be dispensed with. It does not seem to
be the opinion of physicists.