Question

exercise 4.18(2) proves that every longitude and every latitude is a line of curvature of a surface if revolution

EXERCISE 4.23. Let S be the torus obtained by revolving about the axis the circle in the xz-plane with radius 1 centered at (2,0,0). This torus is illustrated in Fig. 4.8. Colored red (respectively green) is the region where 2y4 (respectively r2 +y > 4). Let N be the outward-pointing unit 2- normal field on S. (1) Verify that the unit normal vector to every longitudinal every point is n=-N curve at 4. GEOMETRIC CHARACTERIZATIONS OF GAUSSIAN CURVATURE 213 (2) Verify that the unit normal vector n to every latitudinal curve in the red (respectively green) region at every point makes an acute (respectively obtuse) angle with N. (3) Use Exercise 4.18(2) to conclude that K>0 on the green region and K

Answer 1

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