Multivariable Calculus (Math 214) Assignment

due Monday, November 1, 1999

1. Let P be the point, other than (0, 0, 0), where the twisted cubic

   g(t) = (t, t^{2}, t^{3})

   meets the surface S in space that is defined by the equation

   z x^{3} + x y^{2} + y z = 0 .

   Find the angle between the twisted cubic and the normal to the surface S at the point P.

2. The location at time t of a particle moving in space is given by (the parametric form of a twisted cubic curve):

   {	x = 6 t y = 3 t^{2} z = t^{3}

   Find the following:

   1. the velocity.

   2. the speed.

   3. the unit tangent vector.

   4. the acceleration.

   5. the projection of the acceleration on the unit tangent.

   6. the perpendicular projection of the acceleration on the unit tangent.

   7. the point on the curve corresponding to t = 1.

   8. the unit tangent when t = 1.

   9. the curvature vector when t = 1.

   10. the curvature when t = 1.

   11. the principal unit normal vector when t = 1.

   12. the binormal vector when t = 1.

3. The ``standard'' 3-dimensional sphere (sometimes called the hypersphere) is the subset of 4-dimensional space consisting of all points (x, y, z, w) that satisfy the equation

   x^{2} + y^{2} + z^{2} + w^{2} = 1 .

   The 4-dimensional ball (sometimes called the hyperball) is the subset of 4-dimensional space consisting of all points (x, y, z, w) that satisfy the inequality

   x^{2} + y^{2} + z^{2} + w^{2} <= 1 .

   What type of geometric object is obtained:

   1. by intersecting the hypersphere with a plane through the origin?

   2. by intersecting the hyperball with a plane through the origin?

   3. by intersecting the hypersphere with the plane x + y + z + w = 1 ?

   4. by intersecting the hyperball with the plane x + y + z + w = 1 ?

4. If one views the metallic ``skin'' of an airplane as a surface in space, is it correct to suggest that this surface has a tangent plane at each of its points?