DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

40 Chapter 2. Surfaces: Local Theoryif angles measured in the uv-plane agree with corresponding angles in T P M for all P .Weleave itto the reader to check in Exercise 5 that this is equivalent to the conditions E = G, F =0.Sincewe have[EF]F=G[]x u · x u x u · x vx v · x u x v · x v⎡| |⎢= ⎣ x u x v| |⎤⎥⎦T ⎡| |⎢⎣ x u x v| |⎛⎡⎤⎞([]) x u · x u x u · x v 0EG − F 2 x u · x u x u · x v ⎜⎢⎥⎟= det= det ⎝⎣x v · x u x v · x v 0 ⎦⎠x v · x u x v · x v0 0 1⎛⎡⎤T ⎡ ⎤⎞⎛ ⎡ ⎤⎞| | | | | || | |= det ⎜⎢⎥ ⎢ ⎥⎝⎣x u x v n ⎦ ⎣ x u x v n ⎦⎟⎠ = ⎜ ⎢ ⎥⎟⎝det ⎣ x u x v n ⎦⎠| | | | | || | |which is the square of the volume of the parallelepiped spanned by x u , x v , and n. Since n is a unitvector orthogonal to the plane spanned by x u and x v , this is, in turn, the square of the area of theparallelogram spanned by x u and x v . That is,EG − F 2 = ‖x u × x v ‖ 2 ≠0.We remind the reader that we obtain the surface area of the parametrized surface x: U → M bycalculating the double integral∫∫ √‖x u × x v ‖dudv = EG − F 2 dudv.UU⎤⎥⎦ ,2,EXERCISES 2.11. Derive the formula given in Example 1(e) for the parametrization of the unit sphere.2. Compute I (i.e., E, F , and G) for the following parametrized surfaces.*a. the sphere of radius a: x(u, v) =a(sin u cos v, sin u sin v, cos u)b. the torus: x(u, v) =((a + b cos u) cos v, (a + b cos u) sin v, b sin u) (0