4. For the cylinder with patch (a) Find the normal vector U. (b) The Gauss X(u, v) = (R cos u, R sin u, v), equals 0. map is G(u, v) = U. Find the derivative dG of the Gauss map. (c) Verify that the shape operator S(Xu) = −Vxu U equals 1 Xu and S(Xv) = -VXv U R (d) From the previous part, you know a matrix for the shape operator is s- (- / 0) S= R 1 H = = trace(S). 2 Use this information to find the Gauss curvature K = det(S) and the mean curvature

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