MATH 4250/6250 Final Exam
Wednesday, May 6, 12:00 – 3:00 pm
C. McCrory
1. Compute the tangent vector, normal vector, binormal vector, curvature, and torsion of
the curve
~α(t) = (cos t, sin t, cos 2t)
at t= 0.
2. Suppose the binormal vector of a unit speed space curve ~α(s) makes a constant angle
with a fixed vector ~u, and the curvature κand torsion τare never zero. Prove that the ratio
κ(s)/τ(s) is constant, by filling in the details of the following proof outline:
(i) Show ~
B·~u =a, where ais a constant.
(ii) Show ~
N·~u = 0.
(iii) Show ~u =a~
B+f(s)~
T.
(iv) Take the derivative of (iii) and conclude that κ/τ is constant.
3. Find the equation of the tangent plane of the surface
~x(u, v) = (u, v , u2−v3)
at the point ~x(1,−1).
4. Define the following invariants of surfaces:
(a) shape operator
(b) principal curvatures
(c) Gaussian curvature
(d) mean curvature
5. Compute the principal curvatures and the Gaussian curvature of the surface
~x(u, v) = (u2, eucos v , eusin v),−∞ < u < ∞,0< v < 2π.
6. Compute the shape operator Sof the helicoid
~x(u, v) = (ucos v , u sin v, v),0< u < ∞,0< v < 2π
at the point ~x(1, π). (Find the matrix of Swith respect to the basis ~xu,~xv.)
7. Let ~x(u, v) be a coordinate patch with E= 1 and F= 0. Prove that the u-parameter
curves ~α(u) = ~x(u, v0) (v0constant) are geodesics. (Hint: Compute the partial derivatives
of Eand F.)
8. (a) Give the following definitions:
(i) parallel vector field along a curve ~α on a surface M
(ii) holonomy of a closed curve ~α on a surface M
(b) State the angle excess theorem for a geodesic triangle.