3. (a) Let S be a compact regular surface. Show that there exists a point p on S such that K(p) > 0. (Hint: find a point p € S that maximizes the distance to the origin in R³. Show that K(p) > 0. You may find do Carmo, 1-5, problem 14 useful.) (b) Let S be a regular surface that is diffeomorphic to a torus. Show that S contains points where the Gauss curvature is positive, zero and negative.

3. (a) Let S be a compact regular surface. Show that there exists a point p on S such that K(p) > 0. (Hint: find a point p € S that maximizes the distance to the origin in R³. Show that K(p) > 0. You may find do Carmo, 1-5, problem 14 useful.) (b) Let S be a regular surface that is diffeomorphic to a torus. Show that S contains points where the Gauss curvature is positive, zero and negative.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.6: The Three-dimensional Coordinate System

Problem 41E: Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?

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