Dr. Cleary's Math A6100/461 Homework Assignments:
- HW1, due Feb 5th
- (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.
- (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.
- 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
- HW7, due Wed, Mar 19th
- 5.6.1
- 6.1.1
- 6.1.4
- Show that Mercator's parameterization of the sphere is conformal.
- Consider the surface of revolution obtained by rotating the curve x=x(t) > 0 , z=z(t) around the z-axis.
Comupute its first fundamental form and show that the curves of constant t and theta are always orthogonal. Is this parameterization always a conformal map from the plane?
- HW9, due Wed, Mar 26th
- 7.1.1, 7.1.2
- 7.2.1,7.3.2, 7.3.3, and compute the Gauss map of the hyperboloid of one sheet x^2 + y^2 -z^2 =1
- 7.4.2
- HW10, due Wed Apr 2nd
- 8.1.1, 8.1.2, and compute the Gaussian and mean curvatures of the surface s(u,v)= (u+v, 2u-2v, uv) at the point (4,0,4).
- HW11, due Wed, Apr 9th
- HW12, due Mon, Apr 28th
- 9.1.1,9.1.2 part (i), 9.3.2, 9.4.1
- HW13 due Wed, May 7th
- 10.2.1
- 10.2.2 (note that 9.5.1 ( done in class) and 6.1.4 (done earlier) are useful)
- 10.2.4 (also builds on 9.5.1)
- HW14 due Mon, May 12th (last assignment!)