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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.)