§ 1. Classical Differential Calculus
161
other purpose than to express the coefficients ρ and θ as partial derivatives
of the change of curvilinear coordinate formulas.
(iii) Covariant derivatives on a Cartesian space. A “ differential ” or a
“ covariant derivative ” can be assigned to any C
1 tensor field T . It is also written T or, in classical Riemannian geometry, ∇T , where the sign ∇, “ nabla ”,
supposed to come from Pharaonic Egyptian, must have been chosen by the
inventors of tensor calculus to highlight its esoteric character; this operation
takes a vector field of (p, q) to one of type (p + 1, q). When calculating in a
chart, a first idea that comes to mind is to go, for example, from a tensor
field T of type (2, 1) whose coefficients (5) depend on the coordinates ξ
i of
the point x, to a tensor field of type (3, 1) whose coefficients are reportedly
the functions D i T
h
jk (ξ), where D i = d/dξ
i . This is a bad idea because differentiating formulas (10) would give rise to second derivatives of the ξ with
respect to the η, which would not lead us to the components of a tensor. It
is better, as always, to argue geometrically: since T associates the trilinear
function (k, l, u) → T (x; k, l, u) to every x ∈ U , a quadrilinear function can
be deduced by differentiating with respect to x for given k, l, u ; this is the
covariant derivative T
of the tensor field T defined by the formula
T
(x; h, k, l, u) =
d
dt
T (x + th; k, l, u) for t = 0 ,
(3.11)
= d 1 T [(x; h), k, l, u]
in accordance to (2.22), with h, k, l ∈ E and u ∈ E
∗ . The result is easily
calculated by using a basis (a i ) for E independent of x . Indeed, then
T (x; k, l, u) = T
r
pq (x)k
p l
q u r
with coefficients T
r
pq (x) = T (x; a p , a q , a
r ), and so obviously
T
(x; h, k, l, u) = dT
r
pq (x; h)k
p l
q u r = D i T
r
pq (x)h
i k
p l
q u r ,
where D i = d/dx
i . Therefore, in this case, the coefficients of T
are indeed
the partial derivatives of the coefficients of T with respect to the Cartesian
coordinates of x. But as we want to use the variable basis (a i (ξ)) for all
calculations at the point x = f (ξ), we have to argue differently.
Applying the definition of T
for x = f (ξ), h = a i (ξ), k = a p (ξ), l = a q (ξ)
and u = a
r (ξ), we need to calculate
20
∇ i T
r
pq (ξ) = T
[x; a i (ξ), a p (ξ), a q (ξ), a
r (ξ)] =
(3.12)
=
d
dt
T [x + ta i (ξ); a p (ξ), a q (ξ), a
r (ξ)]
20 The traditional notation ∇iT
r
pq indicates that we can go from the components of
T to those of T
by “ partial covariant differentiations ” similar but not identical
to classical partial differentiations with respect to the variables ξ
i .
161
other purpose than to express the coefficients ρ and θ as partial derivatives
of the change of curvilinear coordinate formulas.
(iii) Covariant derivatives on a Cartesian space. A “ differential ” or a
“ covariant derivative ” can be assigned to any C
1 tensor field T . It is also written T or, in classical Riemannian geometry, ∇T , where the sign ∇, “ nabla ”,
supposed to come from Pharaonic Egyptian, must have been chosen by the
inventors of tensor calculus to highlight its esoteric character; this operation
takes a vector field of (p, q) to one of type (p + 1, q). When calculating in a
chart, a first idea that comes to mind is to go, for example, from a tensor
field T of type (2, 1) whose coefficients (5) depend on the coordinates ξ
i of
the point x, to a tensor field of type (3, 1) whose coefficients are reportedly
the functions D i T
h
jk (ξ), where D i = d/dξ
i . This is a bad idea because differentiating formulas (10) would give rise to second derivatives of the ξ with
respect to the η, which would not lead us to the components of a tensor. It
is better, as always, to argue geometrically: since T associates the trilinear
function (k, l, u) → T (x; k, l, u) to every x ∈ U , a quadrilinear function can
be deduced by differentiating with respect to x for given k, l, u ; this is the
covariant derivative T
of the tensor field T defined by the formula
T
(x; h, k, l, u) =
d
dt
T (x + th; k, l, u) for t = 0 ,
(3.11)
= d 1 T [(x; h), k, l, u]
in accordance to (2.22), with h, k, l ∈ E and u ∈ E
∗ . The result is easily
calculated by using a basis (a i ) for E independent of x . Indeed, then
T (x; k, l, u) = T
r
pq (x)k
p l
q u r
with coefficients T
r
pq (x) = T (x; a p , a q , a
r ), and so obviously
T
(x; h, k, l, u) = dT
r
pq (x; h)k
p l
q u r = D i T
r
pq (x)h
i k
p l
q u r ,
where D i = d/dx
i . Therefore, in this case, the coefficients of T
are indeed
the partial derivatives of the coefficients of T with respect to the Cartesian
coordinates of x. But as we want to use the variable basis (a i (ξ)) for all
calculations at the point x = f (ξ), we have to argue differently.
Applying the definition of T
for x = f (ξ), h = a i (ξ), k = a p (ξ), l = a q (ξ)
and u = a
r (ξ), we need to calculate
20
∇ i T
r
pq (ξ) = T
[x; a i (ξ), a p (ξ), a q (ξ), a
r (ξ)] =
(3.12)
=
d
dt
T [x + ta i (ξ); a p (ξ), a q (ξ), a
r (ξ)]
20 The traditional notation ∇iT
r
pq indicates that we can go from the components of
T to those of T
by “ partial covariant differentiations ” similar but not identical
to classical partial differentiations with respect to the variables ξ
i .