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# hw6 - M421 HW 6 Due Friday Nov 30 From Wade Section 11.4...

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Unformatted text preview: M421 HW 6 Due Friday Nov. 30 From Wade Section 11.4 11.5 11.6 Page Number 350-351 357-358 368 Problems 2, 3, 8 1, 5a, 7 3 Non-book Exercises 1) Let be a curve in R3 , = (t) = (1 (t), 2 (t), 3 (t)) t [0, 1] , where C 2 ([0, 1], R3 ) satisfies (t) = 1 for all t [0, 1]. Suppose that and are C 1 ([0, 1], R3 ) functions which satisfy (t) = (t) = 1. Find a condition on and such that the map F : R3 R3 given by F (t, s1 , s2 ) = (t) + s1 (t) + s2 (t), defines a C 1 coordinate system local to the curve . That is, find conditions which make F invertible in some neighborhood of each point of with F -1 C 1 . 2) Let be a smooth two-dimensional submanifold of R3 , ie. = (t) = 1 (t), 2 (t), 3 (t) where C 2 ([0, 1] [0, 1], R3 ), satisfies t = (t1 , t2 ) [0, 1] [0, 1] , t1 (t) t2 (t) = 1. Let (t) be a normal to at (t) choosen to be locally smooth (there are two normals at each point, choose consistently). Show that the map F (t, s) = (t) + s(t) taking R3 to R3 is locally invertible in a neigborhood of each point (t) with a C 1 inverse. 1 ...
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