Math 227 Problem Set VI
Due Wednesday, March 2
1. Identify (or, failing that, describe) each of the following surfaces. In each case, find two different
parametrizations r(u, v) for the surface.
(a) x2 + y 2 + z 2 = 1, z > 0.
(b) z 2 = x2 + y 2 , z > 0.
2. Identify (or, failing that, describe) each of the following surfaces and find an expression for a
normal vector at each point of the surface.
(a)
x = 3 cos θ sin φ,
y = 2 sin θ sin φ,
z = cos φ,
(b) x = sin v,
y = u,
z = cos v,
(c)
y = (2 − cos v) sin u,
z = sin v,
x = (2 − cos v) cos u,
(d) x = r cos θ,
y = r sin θ,
z = θ,
0 ≤ θ ≤ 2π,
−1 ≤ u ≤ 3,
−π ≤ u ≤ π,
0 < r < 1,
0≤φ≤π
0 ≤ v ≤ 2π
−π ≤ v ≤ π
0 < θ < 4π
3. Given a sphere of radius 2 centred at the origin, find the equation for the plane that is tangent
√
to it at the point (1, 1, 2) by considering the sphere as
(a) a surface parametrized by (θ, φ) 7→ (2 cos θ sin φ, 2 sin θ sin φ, 2 cos φ)
(b) a level surface of f (x, y, z) = x2 + y 2 + z 2
p
(c) the graph of g(x, y) = 4 − x2 − y 2
4. Find the surface area of the torus obtained by rotating the circle (x − R)2 + z 2 = r 2 , y = 0
(the circle is contained in the xz–plane) about the z–axis.
1
,
y 2 +z 2
5. Show that the surface x = √
where 1 ≤ x < ∞ can be filled but not painted!
6. Find the area of that part of the cylinder x2 + y 2 = 2ay lying outside z 2 = x2 + y 2 , by
parametrizing the cylinder using the cylindrical coordinates θ and z.
7. A thin spherical shell of radius a is centred at the origin. Find the centroid, (x̄, ȳ, z̄), of the
part of the sphere that lies in the first octant, by parametrizing the sphere using the spherical
coordinates θ and φ. Here x̄, for example, is the average value of x.