Consider the helicoid parametrized as X(u1, u2) = (u1 cos 3u2, u1 sin 3u2, 5u2), 0 u1 5
Chapter 8, Problem 10(choose chapter or problem)
Consider the helicoid parametrized as X(u1, u2) = (u1 cos 3u2, u1 sin 3u2, 5u2), 0 u1 5, 0 u2 2. Let S denote the underlying surface of the helicoid and let be the orientation 2-form defined in terms of X as X(u1,u2)(a, b) = det 5 sin 3u2 a1 b1 5 cos 3u2 a2 b2 3u1 a3 b3 . (a) Explain why the parametrization X is incompatible with . (b) Modify the parametrization X to one having the same underlying surface S but that is compatible with . (c) Alternatively, modify the orientation 2-form to so that the original parametrization X is compatible with .(d) Calculate S , where = z dx dy (x 2 + y2) dy dz and S is oriented using .
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