162
IX – Multivariate Differential and Integral Calculus
for t = 0. For this, let us differentiate the function
T
r
pq (ξ) = T [f (ξ); a p (ξ), a q (ξ), a
r (ξ)]
with respect to ξ by applying the general formulas (2.25) and (2.27) stated at
the end of n
◦ 2. The result is the sum of fours partial differentials with respect
to the four variables on which T (x, h, k, u) depends, it being understood that,
in these differentials, the functions f (ξ), a p (ξ), . . . and their differentials with
respect ξ to will have to be substituted to x, h, k, u and to their differentials,
as if we had to differentiate the product f (ξ)a p (ξ)a q (ξ)a
r (ξ). Differentiating
T (x; k, h, u) with respect to x, by definition, we get d 1 T [(x; dx), h, k, u] =
T
(x; dx, h, k, u) by (11); the first term of the result sought is, therefore,
T
[f (ξ); df (ξ), a p (ξ), a q (ξ), a
r (ξ)] = T
[f (ξ); f
(ξ)dξ, a p (ξ), a q (ξ), a
r (ξ)] .
In the three other terms, we differentiate a linear function; hence it is sufficient to replace the variable in T , for example, a p (ξ), which depends on ξ,
by its differential da p (ξ; dξ). Replacing dξ by the variable h ∈ R
n on which
the differential of a function of ξ ∈ R
n depends, we, therefore, finally get
dT
r
pq (ξ; h) = T
[x; df (ξ; h), a p (ξ), a q (ξ), a
r (ξ)] +
+ T [x; da p (ξ; h), a q (ξ), a
r (ξ)] +
+ T [x; a p (ξ), da q (ξ; h), a
r (ξ)] +
+ T [x; a p (ξ), a q (ξ), da
r (ξ; h)] .
As we are interested in coefficient (12) of T
, in the above, we need to take
h = e i since df (ξ; h) = a i (ξ) on the left hand side of the previous formula.
da p (ξ; e i ) = D i a p (ξ), etc. remain to be expressed in terms of the a j (ξ) and
a
j (ξ) themselves ; for this, set
D i a p (ξ) = Γ
j
ip (ξ)a j (ξ) , D i a
p (ξ) = Δ
p
ij (ξ)a
j (ξ)
(3.13)
with numerical coefficients to be determined, i.e. the Christoffel symbols. In
view of (12) and the multilinearity of T , we get
D i T
r
pq (ξ) = ∇ i T
r
pq (ξ) + Γ
j
ip (ξ)T
r
jq (ξ) + Γ
j
ir (ξ)T
r
pj (ξ) + Δ
r
ij (ξ)T
j
pr (ξ) ,
and so, omitting the ξ, the formula
∇ i T
r
pq = D i T
r
pq − Γ
j
ip T
r
jq − Γ
j
iq T
r
pj − Δ
r
ij T
j
pq .
(3.14)
Note that, since a j (ξ) = f
(ξ)e j = D j f (ξ),
21
D i a j (ξ) = D i D j f (ξ) = D j D i f (ξ) = D j a i (ξ) ,
21 If f is C
2 , a harmless assumption in a context limited to quasi-formal calculations.
IX – Multivariate Differential and Integral Calculus
for t = 0. For this, let us differentiate the function
T
r
pq (ξ) = T [f (ξ); a p (ξ), a q (ξ), a
r (ξ)]
with respect to ξ by applying the general formulas (2.25) and (2.27) stated at
the end of n
◦ 2. The result is the sum of fours partial differentials with respect
to the four variables on which T (x, h, k, u) depends, it being understood that,
in these differentials, the functions f (ξ), a p (ξ), . . . and their differentials with
respect ξ to will have to be substituted to x, h, k, u and to their differentials,
as if we had to differentiate the product f (ξ)a p (ξ)a q (ξ)a
r (ξ). Differentiating
T (x; k, h, u) with respect to x, by definition, we get d 1 T [(x; dx), h, k, u] =
T
(x; dx, h, k, u) by (11); the first term of the result sought is, therefore,
T
[f (ξ); df (ξ), a p (ξ), a q (ξ), a
r (ξ)] = T
[f (ξ); f
(ξ)dξ, a p (ξ), a q (ξ), a
r (ξ)] .
In the three other terms, we differentiate a linear function; hence it is sufficient to replace the variable in T , for example, a p (ξ), which depends on ξ,
by its differential da p (ξ; dξ). Replacing dξ by the variable h ∈ R
n on which
the differential of a function of ξ ∈ R
n depends, we, therefore, finally get
dT
r
pq (ξ; h) = T
[x; df (ξ; h), a p (ξ), a q (ξ), a
r (ξ)] +
+ T [x; da p (ξ; h), a q (ξ), a
r (ξ)] +
+ T [x; a p (ξ), da q (ξ; h), a
r (ξ)] +
+ T [x; a p (ξ), a q (ξ), da
r (ξ; h)] .
As we are interested in coefficient (12) of T
, in the above, we need to take
h = e i since df (ξ; h) = a i (ξ) on the left hand side of the previous formula.
da p (ξ; e i ) = D i a p (ξ), etc. remain to be expressed in terms of the a j (ξ) and
a
j (ξ) themselves ; for this, set
D i a p (ξ) = Γ
j
ip (ξ)a j (ξ) , D i a
p (ξ) = Δ
p
ij (ξ)a
j (ξ)
(3.13)
with numerical coefficients to be determined, i.e. the Christoffel symbols. In
view of (12) and the multilinearity of T , we get
D i T
r
pq (ξ) = ∇ i T
r
pq (ξ) + Γ
j
ip (ξ)T
r
jq (ξ) + Γ
j
ir (ξ)T
r
pj (ξ) + Δ
r
ij (ξ)T
j
pr (ξ) ,
and so, omitting the ξ, the formula
∇ i T
r
pq = D i T
r
pq − Γ
j
ip T
r
jq − Γ
j
iq T
r
pj − Δ
r
ij T
j
pq .
(3.14)
Note that, since a j (ξ) = f
(ξ)e j = D j f (ξ),
21
D i a j (ξ) = D i D j f (ξ) = D j D i f (ξ) = D j a i (ξ) ,
21 If f is C
2 , a harmless assumption in a context limited to quasi-formal calculations.