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aem570-hw1-2010 - AA/EE/ME 570 Manifolds and Geometry for...

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AA/EE/ME 570: Manifolds and Geometry for Systems and Control Homework #1 Due: Thursday January 14, 11:30am The purpose of these problems is to recall some properties of functions and surfaces from calculus. All problems have equal value. 1. Enneper’s surface. Define a surface in R 3 by, x ( u, v ) = u - u 3 3 + uv 2 , v - v 3 3 + vu 2 , u 2 - v 2 . (a) Using any program of your choice, make a plot of the surface. You should get the result in Fig. 1. Figure 1: Enneper’s surface. (b) Find the inverse mapping ( u, v ) in terms of ( x 1 , x 2 , x 3 ) and the conditions under which it exists. When such inverses exist and are continuous, a mapping is termed one-to-one . (c) Show that for u 2 + v 2 < 3, Enneper’s surface has no self-intersections (meaning that given ( u 1 , v 1 ) 6 = ( u 2 , v 2 ) one must have x 1 6 = x 2 ). Hint: use polar coordinates u = r cos( θ ), v = r sin( θ ), and show that the equality ( x 1 ) 2 + ( x 2 ) 2 + 4 3 ( x 3 ) 2 = 1 9 r 2 (3 + r 2 ) 2 holds, where x 1 , x 2 , x 3 are the coordinate functions of Enneper’s surface in R 3 . Then show that the equality implies that points in the (
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