Homework
M474
Fall 2013
Name:
Score:
Consider the regular surfaces:
S1 := {z = x2 + y 2 }
S2 := {z = x2 − y 2 }
For both S1 and S2 do the following computations.
1. Consider the natural global parameterization that views such surfaces as
graphs of functions (i.e. u = x, v = y).
2. For every point of the surface, compute ϕu , ϕv , N as functions from the
surface to R3 .
3. Compute the metric G.
4. Compute dN|(u,v) as a 3 × 2 matrix. Call this matrix A
5. Compute the matrix that represents dN |(u, v) (as a 2 × 2 matrix) in the
bases ϕu , ϕv . Call this matrix B.
6. Compute the matrix that represents the second fundamental form in two
different ways and show they agree:
(a) Create a 2 × 3 matrix by putting ϕu and ϕv in row form. Then
multiply appropriately with the matrix A.
(b) Multiply appropriately the matrices B and G together.
7. Compute at every point principal directions and principal curvatures.
8. Compute at every point the Gaussian curvature.
9. Try to say (I have no idea if this is going to be easy to do algebraically or
not, but there’s only one way to find out...) which points of the surface
are elliptic, parabolic or hyperbolic.