Geometry Exam
Fall 20111. Let a, b, c > 0 be such that a2 + b2 = c2. Define the parametrized curve (helix) α : R → R3 by α(s) =
³
a cos¡s c
¢, a sin¡s c
¢, b¡s c
¢´, s ∈ R.
(a) (5 points) Show that α is parametrized by arc length.
(b) (10 points) Determine the curvature and the torsion of α at each point in R.
(c) (5 points) Determine the osculating plane of α at each point in R.
[Note: b(s) is normal to the osculating plane at s.]
(d) (5 points) Show that the lines containing n(s) and passing through α(s) meets the z axis at a constant angle of π/2.
(e) (5 points) Show that the tangent lines to α make a constant angle with the z axis.
2. Let f (x, y, z) =¡
x + y + z − 1¢2
(a) (5 points) Locate the critical points and critical values of f.
(b) (5 points) For what values of c is the set f (x, y, z) = c a regular values of f.
3. Let S = {(x, y, z) ∈ R3; z = x2− y2}.
(a) (5 points) Show that S is a regular surface.
(b) (5 points) Show that X(u, v) =¡
u + v, u − v, 4uv¢
, (u, v) ∈ R2, is a parametrization for S.
4. (10 points) Describe the region of the unit sphere covered by the image of the Gauss map of the paraboloid of revolution z = x2+ y2.
5. Let a > r > 0, and T2 denote the torus obtained from rotating the circle ¡
y − a¢2
+ z2 = r2 about z axis, and let
X(u, v) =³¡
a + r cos u¢
cos v, ¡
a + r cos u¢
sin v, r sin u
´
, 0 < u < 2π, 0 < v < 2π, be a parametrization which covers the torus T2 except for a meridian and a parallel.
(a) (10 points) Find the surface area A(T2) of the torus.
(b) (10 points) Find the Gaussian curvature at each X(u, v) ∈ T2.
6. (a) (10 points) Let C ⊂ S be a regular curve on a surface with Gaussian curvature K > 0, Show that the curvature κ of C at p satisfies
κ ≥ min¡
|κ1|, |κ2|¢ where κ1 and κ2 are the principal curvatures of S at p.
(b) (10 points) Show that the mean curvature H at p ∈ S is given by
H = 1 π
Z π
0
κn(θ) dθ
where κn(θ) is the normal curvature at p along the direction making an angle θ with a fixed direction, say e1.
[Hint: For each v ∈ TpS with |v| = 1, we have v = e1cos θ + e2sin θ, where e1, e2 are the principal directions at p, and θ is the angle from e1 to v in the orientation of TpS, and κn(θ) = IIp(v) = − < dNp(v), v > . ]
