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 .

Biology 130 : Exam 3 Diversity of Life Diversity of Life : total of all variations in forms, functions, and behaviors. Biodiversity : number of all the surviving species in the world. Number of species increases due to evolution and decreases due to extinction. CLASSIFICATION OF LIVING ORGANISMS Domain, Kingdom, Phylum, Class, Order, Family, Genus, Species. Living organisms are currently classified in to 3 domains and 6 kingdoms. Kingdoms are Archaebacteria, Eubacteria, Plantae, Animalia, Fungi, Protista. Domain ArchaeaProkaryotic Kingdom Archaebacteria Has many ancestral characteristics –Cell wall type 1 Can survive in extreme conditions Domain BacteriaProkaryotic Kingdom Eubacteria It’s the typical bacteria