FINAL EXAM QUESTIONS FROM PREVIOUS YEARS.
(SOME FROM OPEN BOOK EXAMS.)
Curves.
1. Let f:R→Rbe a differentiable function. Consider the plane curve γ:R→R2, γ(t) =
(t, f (t)). Prove a formula for the curvature of γat each point γ(t0). Use your result to
determine the curvature of the parabolas y=kx2at the origin.
2. Suppose αis a curve in R3parametrized by arclength with curvature k6= 0 and torsion
τ. Show that the curvature of β=α0is given by q1 + τ2
k2.
Tangent space and derivatives of maps.
1. Consider the cylinder S={(x, y, z )∈R3:x2+z2= 1}. Introduce coordinates (θ , z) by
x= cos(θ), y = sin(θ), z =z. For each integer nconsider the map ϕn:S→Swhich in the
above coordinates takes the form (θ, z)7→ (nθ, z ). Determine the differential d(ϕn)(1,0,0) of
ϕnat (1,0,0) and the image of the tangent vector (1,0,1) at this point under d(ϕn)(1,0,0).
General second fundamental form questions.
1. Let A, B and Cbe real numbers. Determine the second fundamental form at (0,0,0) of
the regular surface z=Ax2+Bxy +Cy2in R3, and its Gauss and mean curvature there.
Also express the normal curvature along a direction (x, y, 0) ∈T0Sas a function of A, B, C
and the angle θbetween (1,0,0) and (x, y, 0).
2. Let Sbe a regular surface. Prove that if a point p∈Sis elliptic, then there is a
neighbourhood Uof pin Ssuch that all the points of Uare elliptic. Is the same true for
hyperbolic, parabolic, or planar points? Give examples!
3. At a certain point Pon the surface S, the first and second fundamental forms are:
E= 2, F = 1, G = 1, and e= 4, f = 1, g = 1. Find the angle θin the tangent plane TpS
between xuand the principal direction e1.
Gauss curvature questions.
1. Assume that x(u, v ) parameterizes a regular surface with unit normal N, Gaussian
curvature Kand mean curvature H. Assume furthermore that all the coordinate curves
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