Homework 8
Math 352, Fall 2014
Due Date: Friday, October 31
1. Let H be the hyperplane x1 + x2 + x3 + x4 = 4 in R4 . Find a parametrization for the
(two-dimensional) unit sphere on H centered at the point (1, 1, 1, 1).
2. The paraboloid x3 = x21 + x22 on the x1 x2 x3 -hyperplane in R4 is reflected across the
hyperplane x1 + x2 + x3 + x4 = 1. Find parametric equations for the resulting surface.
3. (a) Find formulas (in terms of u and v) for two perpendicular unit vectors that are
both perpendicular to the vector (cos u sin v, sin u sin v, cos v).
Hint: Use tangent vectors to the unit sphere.
(b) Let M be the 3-manifold in R6 consisting of all points (x1 , x2 , x3 , x4 , x5 , x6 ) for
which (x1 , x2 , x3 ) and (x4 , x5 , x6 ) are perpendicular unit vectors in R3 . Find a
parametrization of M .
4. The equation
p
2
2
x1 + x 2 + x3
2
2
− 2 + x4 2 + x5 2 = 1
defines a “sphere of spheres” in R5 . Find a parametrization of this 4-manifold.
p
2
2
2
Hint: Consider the torus
x + y − 2 + z 2 = 1 in R3 .