244
IX – Multivariate Differential and Integral Calculus
We can now return to the torus T
2 , to the subgroup H of t such that
tu
∈ G
= G ∩ D
and to the subgroup
G = D + G
= D + Z
2 ,
which is the inverse image of Z = f (D) under f .
If H = {0}, then G = D and so Z
2
⊂ D, which is absurd.
If H = R, then G
= D
, so that G contains D and D
, and so G = R
2 ,
i.e. R
2 = D + Z
2 . The set of lines parallel to D is therefore countable ; which
is absurd.
If H is in the case (d), G
= G∩D
is everywhere dense in D
, G = D +G
is everywhere dense in D + D
= R
2 , and as f (R) = F (G), f (R) is clearly
everywhere dense in T
2 . If a/b were rational, the image f (R) = F (G) would
be compact, thus closed, and so F (G) = T
2 , which is absurd.
To conclude, let us study f (R) = F (D) = Z in the neighbourhood of the
identity element (1, 1) of T
2 . First note that the map F from R
2 onto T
2
given by (9) is already surjective on the closed square
K : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1
and that it is injective on the open square. Drawing the verticals with abscissa
p and horizontal ordinates q, where p, q ∈ Z gives a grid of R
2 which cuts
D into intervals that can be numbered by the n ∈ Z ; each of these intervals
can be mapped by an integer translation into a line segment parallel to D
contained in K whose endpoints are on each side of K. Figure 8 is obtained
from the interval K ∩ D.
If the slope of D is rational, these segments are periodically reproduced.
Indeed we saw at the start of the proof that by assuming a and b to be integers
and coprime,
61 relation f (t) = f (t
) is equivalent to t = t
mod Z ; but then
the points tw and t
w of D, where w = (a, b), differ from each other by an
element of Z
2 , so that the intervals of D containing them can be deduced
from each other by an integer translation; by bringing them back into K, we
thereby get the same line segments, and so the announced periodicity holds.
These arguments also prove that all these segments can be obtained by only
considering the intersections of the set of points tw ∈ D with the grid, where
0 ≤ t ≤ 1. There are obviously finitely many of them. The union Δ of these
line segments constructed in K is, therefore, compact, and hence, as seen
earlier, so is the curve F (Δ) of T
2 .
If the slope of D is, on the contrary, irrational, then these segments are
pairwise disjoint, for were it not so, there would be numbers t and t
=
t such that tw − t
w ∈ Z
2 , so that w = (a, b) would be proportional to
an integer vector : so the ratio a/b would be rational.
62 As f (R) = F (D)
61 This means that the vector w = (a, b) generating D is a primitive element of Z
2 .
We use this term for any integral vector belonging to a basis of Z
2 .
62 These arguments show that f is injective if and only if a/b /
∈ Q.
IX – Multivariate Differential and Integral Calculus
We can now return to the torus T
2 , to the subgroup H of t such that
tu
∈ G
= G ∩ D
and to the subgroup
G = D + G
= D + Z
2 ,
which is the inverse image of Z = f (D) under f .
If H = {0}, then G = D and so Z
2
⊂ D, which is absurd.
If H = R, then G
= D
, so that G contains D and D
, and so G = R
2 ,
i.e. R
2 = D + Z
2 . The set of lines parallel to D is therefore countable ; which
is absurd.
If H is in the case (d), G
= G∩D
is everywhere dense in D
, G = D +G
is everywhere dense in D + D
= R
2 , and as f (R) = F (G), f (R) is clearly
everywhere dense in T
2 . If a/b were rational, the image f (R) = F (G) would
be compact, thus closed, and so F (G) = T
2 , which is absurd.
To conclude, let us study f (R) = F (D) = Z in the neighbourhood of the
identity element (1, 1) of T
2 . First note that the map F from R
2 onto T
2
given by (9) is already surjective on the closed square
K : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1
and that it is injective on the open square. Drawing the verticals with abscissa
p and horizontal ordinates q, where p, q ∈ Z gives a grid of R
2 which cuts
D into intervals that can be numbered by the n ∈ Z ; each of these intervals
can be mapped by an integer translation into a line segment parallel to D
contained in K whose endpoints are on each side of K. Figure 8 is obtained
from the interval K ∩ D.
If the slope of D is rational, these segments are periodically reproduced.
Indeed we saw at the start of the proof that by assuming a and b to be integers
and coprime,
61 relation f (t) = f (t
) is equivalent to t = t
mod Z ; but then
the points tw and t
w of D, where w = (a, b), differ from each other by an
element of Z
2 , so that the intervals of D containing them can be deduced
from each other by an integer translation; by bringing them back into K, we
thereby get the same line segments, and so the announced periodicity holds.
These arguments also prove that all these segments can be obtained by only
considering the intersections of the set of points tw ∈ D with the grid, where
0 ≤ t ≤ 1. There are obviously finitely many of them. The union Δ of these
line segments constructed in K is, therefore, compact, and hence, as seen
earlier, so is the curve F (Δ) of T
2 .
If the slope of D is, on the contrary, irrational, then these segments are
pairwise disjoint, for were it not so, there would be numbers t and t
=
t such that tw − t
w ∈ Z
2 , so that w = (a, b) would be proportional to
an integer vector : so the ratio a/b would be rational.
62 As f (R) = F (D)
61 This means that the vector w = (a, b) generating D is a primitive element of Z
2 .
We use this term for any integral vector belonging to a basis of Z
2 .
62 These arguments show that f is injective if and only if a/b /
∈ Q.