Week 45

This week has been devoted to the important concept of Gaussian curvature, measuring how surfaces curve and bend in space.  The main theoretical result is Teorema Egregium, which says that curvature is invariant under isometries.  Next week we shall study the closest analogs of  lines in differential geometry, geodesics, and the rest of the semester we will explore the interactions between geodesics, curvature and the global topology of the surface.

Exercises for Tuesday, Nov 11:  5.4: 5, but replace the parametrization x(u,v) to  (u+cosh(v)i) and y(u,v) to the surface of rotation generated by the curve  (1/cosh(v), v-tanh(v)). (Same surface, but simpler calculations.)       5.5:  1, 2, 3.