PRACTICE MIDTERM 1.
1. (a). Consider the parameterized ellipse
r(t) = (2 cos t, sin t, 0).
Calculate the curvature at t= 0.
(b). What is the torsion?
2. Decide which of the following are regular surfaces and explain your answer.
(a). ex=xyz.
(b). z2=x2+y2.
3. Show that x2+y2−z2= 1 is a regular surface and that
(t, θ)→(p1 + t2cos θ, p1 + t2sin θ , t),0< θ < 2π, −∞ < t < ∞
is a parameterization.
4. Decide whether the map
r: (x, y, z)→Ãx
px2+y2+z2,y
px2+y2+z2,z
px2+y2+z2!
is smooth from the cylinder x2+y2= 1 to the sphere x2+y2+z2= 1. Explain.
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