§ 1. Classical Differential Calculus
153
or, more explicitly,
dg [(x, y, z); (h, k, l)] = g(h, y, z) + g(x, k, z) + g(x, y, l) .
(2.25’)
More specifically, suppose that the variables x, y, z are n×n matrices or linear
operators on a Cartesian space and that g(x, y, z) = xyz ; we then get the
formula
dg = dx.yz + xdy.z + xydz ,
where the order of the terms need to be carefully respected; more explicitly,
dg [(x, y, z); (h, k, l)] = hyz + xkz + xyl
where, like x, y, z, h, k, l are matrices or linear operators ; the spaces E, F, G
in (24) are here identical to the vector space L(M ) of linear maps from M
to itself.
Having done this, we can return to the computation of the differential
of the composite function p(u) = g[x(u), y(u), z(u)] we started with. By the
multivariate chain rule, it can be obtained by replacing in dg, the variable
(x, y, z) and its differential (dx, dy, dz) by their expressions in terms of u ;
and so, without assuming g to be multilinear,
dp(u; du) = d 1 g {[x(u); x
(u)du] , y(u), z(u)} +
+ d 2 g {x(u), [y(u); y
(u)du] , z(u)} +
+ d 3 g {x(u), y(u), [z(u); z
(u)du]} .
Having done that, replace du by h. If g is multilinear, the result simplifies by
(25) :
dp(u; h) = g [x
(u)h, y(u), z(u)] + g [x(u), y
(u)h, z(u)] +
(2.26)
+ g [x(u), y(u), z
(u)h] .
If, moreover, the variable u is real, in which case so is h as well, then, because
of the multilinearity of g, h becomes a common factor, and since dp(u; h) =
p
(u)h we get
p
(u) = g [x
(u), y(u), z(u)] + g [x(u), y
(u), z(u)] +
(2.27)
+ g [x(u), y(u), z
(u)]
as if it was a matter of differentiating a product x(u)y(u)z(u) ; this is the key
point. Besides, if it is a genuine product of functions whose values are, for
example, linear operators or n × n matrices, we retrieve the classical formula
d
du
x(u)y(u)z(u) = x
(u)y(u)z(u) + x(u)y
(u)z(u) , +x(u)y(u)z
(u)
where, once again, the order of the factors is essential.