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Unformatted text preview: Physics 570 Homework No. 2 due Wednesday, 3 February, 2010 1. On a 2dimensional manifold, using Cartesian coordinates { x,y } , we are given the local vector fields e V and e C , with components relative to the standard coordinate basis as V x = xy , V y = y 2 , C x = y , C y = x . a. For each of these vector fields, first determine the curve Γ( η ) for which this vector field is the tangent vector, and normalize constants of integration so that for η = 0 the curve is at the initial point on the manifold, x = 2 ,y = 1. Then produce a sketch of the curve and also a few other adjacent curves, i.e., ones with the same tangent vector field but having an initial point somewhere nearby. Such a set of curves is referred to as a congruence of curves , because they are all “congruent” to each other in the sense of the use of the word in plane geometry. b. If the initial point is not particularly nearby, do the curves still look basically the same? Test this by now choosing an initial point as...
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 Spring '10
 DavidS.King
 Linear Algebra, Manifold, Euclidean space

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