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Smooth - Differential Geometry - Solved Exam

Exams, Computational Geometry

Post: February 18th, 2013
Description
This is the Solved Exam of Differential Geometry which includes Normal Vector, Normal Vector, Binormal Vector, Curvature, Torsion, Binormal Vector, Speed Space Curve etc. Key important points are: Smooth, Bijective, Inverse, Continuous, Two Metrics, Riemannian, Standard Di Erential Structure, Gaussian Curvature, Vector Eld, Satisfying
This is the Solved Exam of Differential Geometry which includes Normal Vector, Normal Vector, Binormal Vector, Curvature, Torsion, Binormal Vector, Speed Space Curve etc. Key important points are: Smooth, Bijective, Inverse, Continuous, Two Metrics, Riemannian, Standard Di Erential Structure, Gaussian Curvature, Vector Eld, Satisfying
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Solutions for Midterm Exam Differential Geometry II April 22, 2010 Professor: Tommy R. Jensen Question 1 Is the following mapping x a proper patch? x : R2 → R3 , (u, v ) → (u3 , u + v 3 , v ). Prove that your answer is correct. Is x a diffeomorphism? It is easy to see that x is injective. The partial velocities are xu = (3u2 , 1, 0) and xv = (0, 3v 2 , 1). They are linearly independent, so x is √ a patch. The inverse function x−1 maps (p1 , p2 , p3 ) to ( 3 p1 , p3 ), which is a continuous function, so x is proper. It is not a diffeomorphism, because √ 3 x is not differentiable at x = 0. Correction. To prove that x−1 is not differentiable, we would have to find a patch y in M so that x−1 (y) : R2 → R2 is not differentiable in the usual sense (Definition 5.1). This will not be possible, because in fact the inverse of x is differentiable, which follows from Theorem 5.4. So the correct answer is that x is actually a diffeomorphism! Question 2 Calculate a param..

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