Problem 1. A surface patch has first and second fundamental forms

respectively. Show that the surface is an open subset of a sphere of radius one.

Problem 2. Suppose that a surface patch σ(u, v) has first and second fundamental forms

respectively, where v > 0.

1) Compute the Christoffel symbols and the Gaussian curvature.

2) Prove that L and N do not depend on u.

3) Prove that

4) Prove that

Problem 3. Show that if a surface patch has first fundamental form. where λ is a smooth function of u and v, its Gaussian curvature K satisfies