§1 Differentiable Mappings

DIFFERENTIABLE MAPPlNGS

Let En and U,,, be Euclidean spaces of points x and u, respectively, with coordinates x1 ,...,x, and u1 ,..., urn.A mapping t,b of an open subset X of En into a “subspace” Y of U, is said to be of class C*, t 2 0, if the mapping of X into Urnwith the same values as t,b is of class C‘, in the sense of Appendix I. A mapping t,b of class C*, is of q < t. For brevity, we term a mapping of class class @ for 0 C* a C*-mapping. A mapping t,b of an arbitrary nonempty subspace X of En into a subspace Y of U,,, is said to be of class C* if t,b admits an extension over an open neighborhood of X in En which is a mapping of class C* into a subspace of U r n . A 1-1 mapping is termed biunique; it may be discontinuous. We shall define a diffeomorphism (abbreviated “diff”) of a nonempty open subset X of En onto a subspace Y of U, in two ways and prove these two definitions equivalent.

<

Dz#s. A biunique mapping t,b : X Y of a nonempty open subset X of En onto a subspace Y of U, will be called a C*-dz# if both t,b and its inverse cp : Y + X are C*-mappings, t > 0. Definition 1.1.

j .

A biunique mapping t,b : X + Y of a nonDefinition 1.2. D#s. empty open subset X of En onto a subspace Y of U, will be called a C*-dz#, t > 0, if t,b is of class C‘, and if its Jacobian vanishes at no point of

x.

There is no assumption that Y is open in either definition. However, the openness of Y follows from the second definition and the classical implicit function theory locally applied. We shall prove the following lemma: 7

8

I.

ANALYSIS OF NONDEGENERATE FUNCTIONS

Lemma 1.1. The above two dejnitions of a day of X onto Y are equivalent. It is trivial that a mapping which is a diff in the sense of Definition 1.2 is a diff in the sense of Definition 1.1. It remains to show that a mapping J/I which satisfies the conditions of Definition 1.1 has a nonvanishing Jacobian at each point of X. If 'p is the inverse of 4, then if x is an arbitrary point of X, ('p

O

M x ) = XI

x E XI

(1.1)

(see Appendix I). By hypothesis, 'p admits an extension 'pe as a mapping of class C1 of an open neighborhood Ye of Y in Uninto En. If one sets

'p"4

= ('pl"~),...,

4(x) = (h(x),...1

'pn8(U)),

cd&)),

l4 E

ye,

x E XI

it follows from the identity (1.1) and the chain rule that at each point E X,for k and p on the range 1,2,...,n

x

where Sku is the Kronecker delta. We are following a convention of tensor algebra whereby the term inscribed on the left of (1,2), with index j repeated in both factors, is summed for j on its range 1, 2,..., n. If Joland J&are the Jacobians of the mappings 'pe and 4, it follows from (1.2) that at each point x E X

JW(4 JdX)

= 1,

24

= #(x).

Hence J4(x) # 0 at each point x E X. Lemma 1.1 follows. A particular consequence of the equivalence of the two definitions is that under the conditions of either definition #(X) is an open subset of U, and that $-l is a diff of $(X) onto X. Let En,Ek ,EL be Euclidean n-spaces. By means of either definition one readily proves the following: If 4 is a diff of an open subset X of Enonto an open subspace Y of Ek and 'p is a diff of Y onto an open subspace Z of E i ,then y o 4,1 is a d 8 of X onto Z.

1.

DIFFERENTIABLE MAPPINGS

9

Definition 1.3. Critical Points. Given a real-valued function x + f ( x ) of class C1 defined in an open subset X of E n , a critical point off is a point x E X at which each of the first-order partial

derivatives off vanishes. A point of X which is not critical is termed ordinary.