PROBLEM SET VII
DUE FRIDAY,  APRIL
De nition. Consider the sphere
S2 := {(x, y, z, ) ∈ A3 | x2 + y2 + z2 = 1}
and the cylinder
Z2 := {(x, y, z) ∈ A3 | x2 + y2 = 1}.
Write p : [0, 1] .
R for the function
√
p(z) := 1 − z2 ,
and choose an increasing homeomorphism g : (0, π) .
sponding map
Mg : S2 − {(0, 0, −1), (0, 0, 1)} .
(
as follows.
Mg (x, y, z) :=
R. De ne a correZ2
)
x
y
,
, g(arccos(z)) .
p(z) p(z)
Exercise . Show that Mg is a homeomorphism. Write a formula for the inverse M−1
g .
Exercise . Show that if g = cot, then the unique ray starting at the origin
and passing through (x, y, z) ∈ S2 intersects Z2 at Mg .
Exercise . Recall the inverse Gudermannian gd−1 : (−π/2, π/2) .
last semester:
∫ s
−1
gd (s) :=
sec .
R from
0
−1
Let g(s) = gd (s−π/2). For any real number α, consider the curve γ : R . Z2
given by γ(t) := (cos(t), sin(t), αt). Suppose h ∈ (−1, 1). At what angle does
the curve γ intersect the circle
{(x, y, z) ∈ Z2 | z = h}
At what angle does the curve λ :=
M−1
g
?
◦ γ intersect the circle
{(x, y, z) ∈ S2 | z = h} ?


DUE FRIDAY,  APRIL
Exercise . Consider the n-dimensional disk
{
Dn :=
(x1 , . . . , xn ) ∈ An |
n
∑
}
x2i = 1 .
i=1
Compute the limit
lim vol(Dn )
n→∞
of the volumes of the disks Dn . (Hint: you don’t actually have to nd a formula
for vol(Dn ) to complete this problem.)