154
IX – Multivariate Differential and Integral Calculus
(iv) Diffeomorphisms. Let U be an open subset of a Cartesian space E and
f a map from U to a Cartesian space F of the same dimension as E. Suppose
that f maps U on an open subset of F . f is said to be a diffeomorphism of
class C
p from U to V if f as well as the inverse map g = f
−1 : V −→ U are
bijective and of class C
p . Clearly, for y = f (x), the linear maps f
(x) and
g
(y) are mutually inverse. If E = F , then
J g (y)J f (x) = 1 for y = f (x) , g = f
−1
(2.28)
follows and in particular, J f (x) = 0 for x ∈ U .
Conversely, let us start with a C
p map f from an open subset U of E
to F , with dim(E) = dim(F ), and suppose that f
(x) is invertible for all
x ∈ U . The local inversion theorem (Chap. III, § 5, n
◦ 24, Theorem 24),
whose proof in dimension n is similar to the one in dimension 2, tells us that
for all x ∈ U , there is an open neighbourhood U (x) of x homeomorphically
mapped by f onto an open neighbourhood V of y = f (x), the inverse map
from V (y) to U (x) also being C
p . If that is the case for all x ∈ U , then
the image V = f (U ) is open, and more generally so is the image of any open
subset of U . If, moreover, f is injective not only in the neighbourhood of each
point, but globally, and so is a bijection from U onto V , then the inverse map
f
−1 : V −→ U can be considered ; it is C
p , so that f is a diffeomorphism.
When we will prove the change of variable formula for a multiple integral,
we will need to consider a bounded open subset U of a Cartesian space E,
its compact closure A and a map f from A to E or, more generally, to a
Cartesian space F . f will be said to be of class C
p on A if f is of class C
p on
U and if the partial derivatives of order ≤ p of the f
j extend to continuous
functions on A. Since A is compact, this is the case if and only if these
derivatives are uniformly continuous on U (Chap. V, § 2, n
◦ 8, corollary 2
of Theorem 8 generalized to n variables). So the traditional notation for
extensions to A of partial derivatives will continue to be used for the vector
D i f (a) with coordinates D i f
j (a), for the linear maps f
(a) : h → D i f (a)h
i
and the Jacobian J f (a) will be defined in the obvious manner for all a ∈ A.
When dim E = dim F , f will be said to be a diffeomorphism of class C
p
from A onto B = f (A) if, moreover, (i) f is bijective, in which case f is
a homeomorphism from A onto the compact set B = f (A), (Chap. III, § 3,
n
◦ 1, Theorems 11 and 12), (ii) f is a C
p diffeomorphism from U onto an
open set V ⊂ F , and so B = ¯
V , (iii) the inverse map f
−1 : B −→ A is C
p in
the sense defined above.
Conditions (ii) and (iii) require
J f (a) = 0 for all a ∈ A
(2.29)
since the two sides of (28) are continuous functions on A. Conversely, if
(29) holds, (ii) follows by (i) and the local inversion theorem, and (iii) is a
consequence of the fact that, if M f (x) is the Jacobian matrix of f at x ∈ U ,
the entries of its inverse, i.e. the derivatives of f
−1 , are the quotients of
IX – Multivariate Differential and Integral Calculus
(iv) Diffeomorphisms. Let U be an open subset of a Cartesian space E and
f a map from U to a Cartesian space F of the same dimension as E. Suppose
that f maps U on an open subset of F . f is said to be a diffeomorphism of
class C
p from U to V if f as well as the inverse map g = f
−1 : V −→ U are
bijective and of class C
p . Clearly, for y = f (x), the linear maps f
(x) and
g
(y) are mutually inverse. If E = F , then
J g (y)J f (x) = 1 for y = f (x) , g = f
−1
(2.28)
follows and in particular, J f (x) = 0 for x ∈ U .
Conversely, let us start with a C
p map f from an open subset U of E
to F , with dim(E) = dim(F ), and suppose that f
(x) is invertible for all
x ∈ U . The local inversion theorem (Chap. III, § 5, n
◦ 24, Theorem 24),
whose proof in dimension n is similar to the one in dimension 2, tells us that
for all x ∈ U , there is an open neighbourhood U (x) of x homeomorphically
mapped by f onto an open neighbourhood V of y = f (x), the inverse map
from V (y) to U (x) also being C
p . If that is the case for all x ∈ U , then
the image V = f (U ) is open, and more generally so is the image of any open
subset of U . If, moreover, f is injective not only in the neighbourhood of each
point, but globally, and so is a bijection from U onto V , then the inverse map
f
−1 : V −→ U can be considered ; it is C
p , so that f is a diffeomorphism.
When we will prove the change of variable formula for a multiple integral,
we will need to consider a bounded open subset U of a Cartesian space E,
its compact closure A and a map f from A to E or, more generally, to a
Cartesian space F . f will be said to be of class C
p on A if f is of class C
p on
U and if the partial derivatives of order ≤ p of the f
j extend to continuous
functions on A. Since A is compact, this is the case if and only if these
derivatives are uniformly continuous on U (Chap. V, § 2, n
◦ 8, corollary 2
of Theorem 8 generalized to n variables). So the traditional notation for
extensions to A of partial derivatives will continue to be used for the vector
D i f (a) with coordinates D i f
j (a), for the linear maps f
(a) : h → D i f (a)h
i
and the Jacobian J f (a) will be defined in the obvious manner for all a ∈ A.
When dim E = dim F , f will be said to be a diffeomorphism of class C
p
from A onto B = f (A) if, moreover, (i) f is bijective, in which case f is
a homeomorphism from A onto the compact set B = f (A), (Chap. III, § 3,
n
◦ 1, Theorems 11 and 12), (ii) f is a C
p diffeomorphism from U onto an
open set V ⊂ F , and so B = ¯
V , (iii) the inverse map f
−1 : B −→ A is C
p in
the sense defined above.
Conditions (ii) and (iii) require
J f (a) = 0 for all a ∈ A
(2.29)
since the two sides of (28) are continuous functions on A. Conversely, if
(29) holds, (ii) follows by (i) and the local inversion theorem, and (iii) is a
consequence of the fact that, if M f (x) is the Jacobian matrix of f at x ∈ U ,
the entries of its inverse, i.e. the derivatives of f
−1 , are the quotients of