Dr. Cleary's Math A6100/461 Homework Assignments:

• HW1, due Feb 5th
  1. (basically 1.1.7 from Pressley) Parameterize a cycloid: the path of a point on the circumference of a circle of radius a as it rolls without slipping along the positive x-axis.
  2. (basically 1.1.7 from Pressley, continued) Parameterize the path of a point at distance b from the center of disk of radius a as it rolls without slipping along the positive x-axis.
  3. Parameterize the path of a point on the circumference of a circle of radius a as it rolls without slipping around the outside of another circle of radius a. You may want to experiment with coins to understand the behavior before parameterizing.
• HW2, due Feb 10th
  • Problems 1.2.1, 1.3.2, 1.3.3 and 1.4.3 from Pressley
• HW3, due Feb 19th
  • Find the curvature of the curve paramtererized as x(t)=t,y(t)=t,z(t)=1+t^2.
  • 2.1.1 #iii, #iv
  • 2.2.6
  • 2.3.1 (missing comma in (i) separating the x and y components)
  • 2.3.2
• HW4, due Feb 26th
  • Find the curvature of the curve paramtererized as x(s)=cos(s)/sqrt(2),y(s)=sin(s),z(s)= cos(s)/sqrt(2) and identify the curve.
  • Spinning around a curve: If a rigid body moves along a unit speed curve gamma (s), we can describe its motion as a translation along s and a rotation around the curve. This rotation is determined by an angular momentum vector w which satisfies T' = w x T, N' = w x N and B' = w x B. Show that w= tau T + kappa B by expressing w in terms of the basis (T,N,B) and solving.
  • Compute T, N and B for the curve x(t)= e^t cos(t), y(t)=e^t sin(t), z(t)=e^t, and the curvature and torision of the curve.
  • Use Green's Theorem to find the area of the ellipse with axes of lengths 2p and 2q and then do 3.2.2
• HW5, due Wed, Mar 5th (day of Exam 1)
  • Use the stereographic projection method descibed in class to give two coordinate charts for the unit sphere.
  • 4.1.2- don't actually do the problem, just find the transition function sigma z + to sigma y -.
  • 4.1.3
  • 4.2.6
• HW6, due Wed, Mar 12th
  • 4.4.1, 4.5.1
  • Which non-degenerate quadric surfaces are a)generalized cylinders? b)generalized cones? c)ruled surfaces? d)surfaces of revolution?
  • 5.2.3

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