288
III - Convergence: Continuous variables
- which clearly will not be enough - on the open set U where it is defined.
For (a, b) given, consider again, this time in the context where Euler used it,
the expression
(23.2)
[f(a + u, b + v) - f(a + u, b)J - [f(a, b + v) - f(a, b)J
where lui and Ivl remain so small that, in all that follows, one needs to
consider values of f only on a compact disc K c U with centre (a, b). For a,
b, u and v given, (2) can be written g(1) - g(O), where
(23.3)
g(t) = f(a + tu, b + v) - f(a + tu, b).
By the results of nO 16
(23.4)
Ig(1) - g(O) - g'(O)1 ::; sup Ig'(t) - g'(O)1
where the sup is taken over t E [0, 1J. But (chain rule)
g'(t)
Dd(a + tu, b + v)u - Dd(a + tu, b)u =
[Dd(a + tu, b + v) - Dd(a, b)Ju - [Dd(a + tu, b) - Dd(a, b)Ju.
If one assumes that Dd is differentiable at the point (a, b), then the first
difference between [ J is, by definition, equal to
DIDd(a, b)tu + D2Dd(a, b)v + o(ltul + IV!)j
the second difference between [ J is similarly equal to
DIDd(a, b)tu + o(ltu!).
So we have
g'(t) = D2Dd(a, b)uv + o(ltul + Iv!)u,
and in particular
g'(O) = D2Dd(a,b)uv + o(lvl)u,
whence Ig'(t) - g'(O)1 = o(ltul + Iv!)u. Since
(Itul + Ivl)lul ::; (lui + Ivl)2,
it follows that
sup Ig'(t) - g'(O)1 = o[(lul + Ivl)2].
The relation (4) can then be written
g(1) - g(O)
[Dd(a, b + v) - Dd(a, b)]u + o[(lul + Iv!)2] =
= [D2Dd(a, b)v + o(v)]u + o[(lul + Iv!)2]j