verify the vector in normal to S at the point f(u,0)

7. Let r: [0, 24] - R3 denote an arc length parameterization of a smooth, simple closed curve C. Assume r"(u) * 0 for every u E [0, 2x]. Notice that r(27) = r(0), r'(2x) = r'(0), r"(2x) = r"(0). Suppose there exists e > 0 such that for any integer k, the curve C can be "thickened" to a surface S in R* with parameterization f : [0, 2x] x (-6, () - R3 given by f(u, v) = r(u) + (vcos ku )r"(u) + (usin )r'(u) x r"(u). (a) (1 point) Verify that f(2x, v) = f(0, (-1)*v) for every ve (-E, E). (b) (1 point) Use the fact that (r'(u)| = 1 to show that r'(u) . r"(u) = 0.[(3) [2 points) Verify that for each 1.1 E [l], 23f] the vector WELL} := {005 %]r'[u} x {'11:} {sin %}r"[u} is normal tn S at the paint u, } = 11:11.} E C. Continued on the next page. Continued fmm the previous page. (d) (2 point} Compute Iw(u]|2 and conclude that w{u} :, 0 for emery 1.1 E [0,231. (e) (1 point) Verify that w(27) = (-1)*w(0). (f) (1 point) S is orientable in the special case where k = 1. O True O False