152
IX – Multivariate Differential and Integral Calculus
E × F × G. To compute dp, introduce the partial differentials of the function
g(x, y, z) with respect to x, y and z. The partial differential d 1 g[(x; h), y, z],
which depends linearly on an additional variable h ∈ E, is obtained by fixing
y ∈ V and z ∈ W and by differentiating the map x → g(x, y, z) ; hence, by
definition,
d 1 g [(x; h), y, z] =
d
dt
g(x + th, y, z) for t = 0
(2.22)
or, in Leibniz style,
d 1 g [(x; dx), y, z] = g(x + dx, y, z) − g(x, y, z) .
(2.22’)
The differentials d 2 g[x, (y; k), z] and d 3 g[x, y, (z; l)] are similarly defined, the
letters k and l denoting vector variables in F and G. As g is defined on an open
subset of E ×F ×G, its (total) differential depends on a vector varying in this
space, i.e. on three vectors h ∈ E, k ∈ F , l ∈ G. By definition, it is obtained by
considering the value of g at the point (x, y, z)+t(h, k, l) = (x+th, y+tk, z+tl)
and by differentiating the result at t = 0 :
dg [(x, y, z); (h, k, l)] =
d
dt
g(x + th, y + tk, z + tl) for t = 0
(2.23)
dg [(x, y, z); (h, k, l)] = d 1 g [(x; h), y, z] + d 2 g [x, (y; k), z] +
(2.24)
+ d 3 g [x, y, (z; l)]
is easily seen to follow. The left hand side is indeed a linear function of the
vector (h, k, l) ∈ E × F × G ; as
(h, k, l) = (h, 0, 0) + (0, k, 0) + (0, 0, l) ,
it is, therefore, equal to dg[(x, y, z); (h, 0, 0)]+ etc. But, by definition,
dg [(x, y, z); (h, 0, 0)] =
d
dt
g [(x, y, z) + t(h, 0, 0)] =
=
d
dt
g(x + th, y, z) for t = 0 ,
an expression equal to d 1 g[(x; h), y, z)] by definition. This gives (24). As an
aside, the following mistake should be avoided : the left hand side of (24) is a
linear function of the vector (h, k, l) in the vector space E × F × G, and not
a trilinear function of the vectors h ∈ E, k ∈ F , l ∈ G.
For example, suppose that, as in the case of a tensor, g(x, y, z) is a trilinear
function of x, y, z. Since x → g(x, y, z) is linear, by (3), its differential d 1 g is
just dx → g(dx, y, z). Hence
dg = g(dx, y, z) + g(x, dy, z) + g(x, y, dz)
(2.25)
IX – Multivariate Differential and Integral Calculus
E × F × G. To compute dp, introduce the partial differentials of the function
g(x, y, z) with respect to x, y and z. The partial differential d 1 g[(x; h), y, z],
which depends linearly on an additional variable h ∈ E, is obtained by fixing
y ∈ V and z ∈ W and by differentiating the map x → g(x, y, z) ; hence, by
definition,
d 1 g [(x; h), y, z] =
d
dt
g(x + th, y, z) for t = 0
(2.22)
or, in Leibniz style,
d 1 g [(x; dx), y, z] = g(x + dx, y, z) − g(x, y, z) .
(2.22’)
The differentials d 2 g[x, (y; k), z] and d 3 g[x, y, (z; l)] are similarly defined, the
letters k and l denoting vector variables in F and G. As g is defined on an open
subset of E ×F ×G, its (total) differential depends on a vector varying in this
space, i.e. on three vectors h ∈ E, k ∈ F , l ∈ G. By definition, it is obtained by
considering the value of g at the point (x, y, z)+t(h, k, l) = (x+th, y+tk, z+tl)
and by differentiating the result at t = 0 :
dg [(x, y, z); (h, k, l)] =
d
dt
g(x + th, y + tk, z + tl) for t = 0
(2.23)
dg [(x, y, z); (h, k, l)] = d 1 g [(x; h), y, z] + d 2 g [x, (y; k), z] +
(2.24)
+ d 3 g [x, y, (z; l)]
is easily seen to follow. The left hand side is indeed a linear function of the
vector (h, k, l) ∈ E × F × G ; as
(h, k, l) = (h, 0, 0) + (0, k, 0) + (0, 0, l) ,
it is, therefore, equal to dg[(x, y, z); (h, 0, 0)]+ etc. But, by definition,
dg [(x, y, z); (h, 0, 0)] =
d
dt
g [(x, y, z) + t(h, 0, 0)] =
=
d
dt
g(x + th, y, z) for t = 0 ,
an expression equal to d 1 g[(x; h), y, z)] by definition. This gives (24). As an
aside, the following mistake should be avoided : the left hand side of (24) is a
linear function of the vector (h, k, l) in the vector space E × F × G, and not
a trilinear function of the vectors h ∈ E, k ∈ F , l ∈ G.
For example, suppose that, as in the case of a tensor, g(x, y, z) is a trilinear
function of x, y, z. Since x → g(x, y, z) is linear, by (3), its differential d 1 g is
just dx → g(dx, y, z). Hence
dg = g(dx, y, z) + g(x, dy, z) + g(x, y, dz)
(2.25)