§5. Differentiable functions of several variables
289
since /uv/ :::; (/u/ 2 + /v/ 2 )/2 we get
(23.5)
But instead of calculating with the function (3), one may also, if Dd is
differentiable at (a, b), use the function
h(t) = f(a + u, b + tv) - f(a, b + tv).
Then one finds
(23.6)
Comparing (5) and (6), we see that
[D2Dd(a,b) - DIDd(a,b)]uv = o[(/u/ + /v/)2];
one can apply this result to u = tuo, v = tvo where Uo and Vo are nonzero
constants and where t tends to 0; then one finds that
[D2Dd(a, b) - D1Dd(a, b)]uovot2 = (/uo/ 2 + /vo/ 2 ) 0(t 2 ),
and deduces the interchangeability of the order of differentiation on dividing
the two sides by t 2 and making t to 0, which yields a right hand side tending
to 0, qed.
The hypotheses used in the proof46, namely that the functions Dd and
D21 are differentiable at (a, b), are in particular satisfied if I is of class C 2 ,
by far the most important case in applications. In general, the subtleties that
one meets in the case of a single real variable rarely generalise, or, when
they do, often involve far too great a cost/efficiency ratio to motivate the
mathematicians. The important problems posed in the theory of differentiable
functions of several variables (differential topology for example) are of another
nature.
The most obvious consequence - and the most important - of (1) is that,
il I is of class en, n ~ 2, then every partial derivative 01 I of order:::; n
can be written in the form Dr DU with p + q :::; n; it is unnecessary to write
expressions such as D~D~DID~DU.
Another consequence: il a function I of class coo is holomorphic, then so
similarly are its successive derivatives I', I" = (I')', etc., which are simply
DIl, DU, etc. So in this case
Dr DU = i q I(p+q) .
Since we shall eventually learn that a holomorphic function is analytic, these
results are not particularly sensational.
.. which I have borrowed from Dieudonne's Treatise on Analysis, Vol. 1, Chap. VIII,
nO 12; he never cites his sources, in this case W.H. Young, beginning of the XX th
century. It goes without saying that, in Dieudonne, one is working in Banach
spaces, which, strictly speaking, does not change the proof at all. You will find
a very different proof in Chap. V, nO 12, Theorem 14; but this uses the integral
calculus and assumes the existence and continuity of D2Dd and D1 D 2/.
289
since /uv/ :::; (/u/ 2 + /v/ 2 )/2 we get
(23.5)
But instead of calculating with the function (3), one may also, if Dd is
differentiable at (a, b), use the function
h(t) = f(a + u, b + tv) - f(a, b + tv).
Then one finds
(23.6)
Comparing (5) and (6), we see that
[D2Dd(a,b) - DIDd(a,b)]uv = o[(/u/ + /v/)2];
one can apply this result to u = tuo, v = tvo where Uo and Vo are nonzero
constants and where t tends to 0; then one finds that
[D2Dd(a, b) - D1Dd(a, b)]uovot2 = (/uo/ 2 + /vo/ 2 ) 0(t 2 ),
and deduces the interchangeability of the order of differentiation on dividing
the two sides by t 2 and making t to 0, which yields a right hand side tending
to 0, qed.
The hypotheses used in the proof46, namely that the functions Dd and
D21 are differentiable at (a, b), are in particular satisfied if I is of class C 2 ,
by far the most important case in applications. In general, the subtleties that
one meets in the case of a single real variable rarely generalise, or, when
they do, often involve far too great a cost/efficiency ratio to motivate the
mathematicians. The important problems posed in the theory of differentiable
functions of several variables (differential topology for example) are of another
nature.
The most obvious consequence - and the most important - of (1) is that,
il I is of class en, n ~ 2, then every partial derivative 01 I of order:::; n
can be written in the form Dr DU with p + q :::; n; it is unnecessary to write
expressions such as D~D~DID~DU.
Another consequence: il a function I of class coo is holomorphic, then so
similarly are its successive derivatives I', I" = (I')', etc., which are simply
DIl, DU, etc. So in this case
Dr DU = i q I(p+q) .
Since we shall eventually learn that a holomorphic function is analytic, these
results are not particularly sensational.
.. which I have borrowed from Dieudonne's Treatise on Analysis, Vol. 1, Chap. VIII,
nO 12; he never cites his sources, in this case W.H. Young, beginning of the XX th
century. It goes without saying that, in Dieudonne, one is working in Banach
spaces, which, strictly speaking, does not change the proof at all. You will find
a very different proof in Chap. V, nO 12, Theorem 14; but this uses the integral
calculus and assumes the existence and continuity of D2Dd and D1 D 2/.