Reading: Chapter 7 through page 137.
Problems: Chapter 7: 7, 8 (Finding the first fundamental form means finding its component functions, also known as metric coefficients.), 9 (The graph is revolved about the u-axis. The function is assumed to be positive. The problem refers to Example (4) on page 119.)
Problem I: Find the equation of the tangent plane at point (x0, y0, z0) to the unit sphere centered at the origin. Use any of your favorite charts for the sphere and assume the point (x0, y0, z0) is covered by your chart.
Problem J: Find the first fundamental form of a plane given by parametrization F(u,v) = p + u q + v r for fixed vectors p, q, and r.
Problem K: Let S be a regular surface in R3 and N: S → S2 an arbitrary unit normal vector field, i.e., a function that assigns each point on S a unit normal vector to the surface S. Show that N is continuous, if and only if N is smooth (differentiable in the terminology of the text). You may assume that (u,v) ↦ xu × xv/||xu × xv|| is smooth as a function from the surface to S2.
Last modified: (2013-02-20 14:25:32 CST)