208
IX – Multivariate Differential and Integral Calculus
But let us consider the linear operator w σ which, for a given permutation σ of {1, . . . , n}, transforms e i into e σ(i) . Then w σ ∈ K and w σ tw
−1
σ =
(t σ(1) , . . . , t σ(n) ). As Δ(w σ tw
−1
σ ) = Δ(t), s 1 = . . . = s n = s by lemma e, and
so
Δ(t) = det(t)
s .
(10.8)
For arbitrary g ∈ G, formula g = utv then shows that
Δ(g) = | det(g)|
s
(10.9)
with an absolute value since Δ(g) must be > 0. This is the general form of
continuous homomorphisms from GL n (R) to R
∗
+ .
To finish the proof of the change of variable formula, it remains to observe
that, if g(x) = tx where t ∈ R is non-zero, then the factor Δ(g) of lemma b
is clearly equal to |t|
−n , and so s = −1.
Exercise. Show that any g ∈ GL n (R) can be written as g = khu where k
is orthogonal, h positive diagonal and u triangular with diagonal (1, . . . , 1) ;
verify (6) for g = u by changing variables in the simple integrals and deduce
(6) for g. (The decomposition g = khu means that any basis (a i ) of R
n can
orthonormalized by a triangular linear map applied to the a i : Gram-Schmidt
orthogonalization process.
Exercise. Set G = GL n (C). Show that any continuous homomorphism
from G to C
∗ (resp. R
∗ ) is of the form g → | det(g)|
s det(g)
p with s ∈ C and
p ∈ Z [resp. s ∈ R, p ∈ {1, −1}].
(ii) Approximation Lemmas. We now return to the general case of the theorem. To prove it, we need some preliminary results justifying what physicists
take to be obvious, that in the neighbourhood of a point a, a diffeomorphism
is approximately linear, and “ so ” multiplies the volumes by the absolute
value of the determinant of its differential map. It then suffices to add the
results and one easily obtains the general formula. . .
In what follows, the length or the norm |u| of a vector u ∈ R
n is defined
by
|u| = sup
|u
1
|, . . . , |u
n
|
;
(10.10)
for this “ cubic ” norm, an open ball centered at a and of radius r is the set
|u
i
− a
i
| < r. To avoid confusion, it will be call the cube centered at a and of
radius radius r, and will be written U (a, r) or K(a, r) according to whether
the cube is open or closed. Because of this norm, a reasonable set can be
approximately decomposed into parallelepipeds whose pairwise intersections
are faces that are negligible in the integration. This operation cannot be
performed using real Euclidean balls. As in any normed vector space, the
norm of a linear map A is defined by
A = sup |Ax|/|x| .
IX – Multivariate Differential and Integral Calculus
But let us consider the linear operator w σ which, for a given permutation σ of {1, . . . , n}, transforms e i into e σ(i) . Then w σ ∈ K and w σ tw
−1
σ =
(t σ(1) , . . . , t σ(n) ). As Δ(w σ tw
−1
σ ) = Δ(t), s 1 = . . . = s n = s by lemma e, and
so
Δ(t) = det(t)
s .
(10.8)
For arbitrary g ∈ G, formula g = utv then shows that
Δ(g) = | det(g)|
s
(10.9)
with an absolute value since Δ(g) must be > 0. This is the general form of
continuous homomorphisms from GL n (R) to R
∗
+ .
To finish the proof of the change of variable formula, it remains to observe
that, if g(x) = tx where t ∈ R is non-zero, then the factor Δ(g) of lemma b
is clearly equal to |t|
−n , and so s = −1.
Exercise. Show that any g ∈ GL n (R) can be written as g = khu where k
is orthogonal, h positive diagonal and u triangular with diagonal (1, . . . , 1) ;
verify (6) for g = u by changing variables in the simple integrals and deduce
(6) for g. (The decomposition g = khu means that any basis (a i ) of R
n can
orthonormalized by a triangular linear map applied to the a i : Gram-Schmidt
orthogonalization process.
Exercise. Set G = GL n (C). Show that any continuous homomorphism
from G to C
∗ (resp. R
∗ ) is of the form g → | det(g)|
s det(g)
p with s ∈ C and
p ∈ Z [resp. s ∈ R, p ∈ {1, −1}].
(ii) Approximation Lemmas. We now return to the general case of the theorem. To prove it, we need some preliminary results justifying what physicists
take to be obvious, that in the neighbourhood of a point a, a diffeomorphism
is approximately linear, and “ so ” multiplies the volumes by the absolute
value of the determinant of its differential map. It then suffices to add the
results and one easily obtains the general formula. . .
In what follows, the length or the norm |u| of a vector u ∈ R
n is defined
by
|u| = sup
|u
1
|, . . . , |u
n
|
;
(10.10)
for this “ cubic ” norm, an open ball centered at a and of radius r is the set
|u
i
− a
i
| < r. To avoid confusion, it will be call the cube centered at a and of
radius radius r, and will be written U (a, r) or K(a, r) according to whether
the cube is open or closed. Because of this norm, a reasonable set can be
approximately decomposed into parallelepipeds whose pairwise intersections
are faces that are negligible in the integration. This operation cannot be
performed using real Euclidean balls. As in any normed vector space, the
norm of a linear map A is defined by
A = sup |Ax|/|x| .