MIDTERM FOR GEOMETRY
Date: Wednesday, April 25, 2001 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1. Consider the curve r(t) = ti + t2j + t3k.
(i) [5%] Find the unit tangent vector t at (0, 0, 0).
(ii) [5%] Find an equation of the osculating plane at (0, 0, 0).
2.
(i) [5%] Is it possible that a differentiable curve whose curvature is zero in some interval but its torsion is not zero in that interval? Why or Why not? On the other hand, is it possible that a differentiable curve whose torsion is zero in some interval but its curvature is not zero in that interval? Why or Why not?
(ii) [5%] Give examples of two curves with the same curvature but different torsion in some interval.
3. Knowing that g11= 1, g12= g21= 0 and g22= cos2(u1). Compute:
(i) [5%] gijgjk (ii) [5%]¡ ∂
∂ujgkl
¢gjk
4. [10%] Let f and h be two differentiable functions of one variable. Compute the first fundamental form of the surface of revolution:
x = f (u) cos v, y = f (u) sin v, z = h(u).
5. [10%] Compute the area of the helicoid
x = u cos v, y = u sin v, z = 2v for 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2π.
6. Let the helicoid be as in problem 5. Compute:
(i) [5%] b12at (1, 0, 0).
(ii) [5%] Γ122at (1, 0, 0).
7. [10%] Let the helicoid be as in problem 5. Find an equation of the tangent plane at (1, 0, 4π) 8. [10%] Let r(u1, u2) be a regular surface. Let m be the unit normal vector of the surface. Show that mi can be written as a linear combination of r1 and r2.
9.
(i) [5%] What is the Gaussian curvature K at the point (0, 0, 1) on the surface x2+ y2+ z2= 1?
(ii) [5%] What is the Gaussian curvature K at the point (√ 3,√
3,√
3) on the surface x2+y2+z2= 9?
10. [10%] Give an example of a differentiable curve whose curvature (as a function of a parameter) can take any positive real values.
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