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Differentiable Function - 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: Differentiable Function, Plane Curve, Curvature, Parabolas, Parametrized, Arclength, Curvature, Tangent Space, Derivatives of Maps, General Second Fundamental
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: Differentiable Function, Plane Curve, Curvature, Parabolas, Parametrized, Arclength, Curvature, Tangent Space, Derivatives of Maps, General Second Fundamental
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SOLUTIONS TO FINAL EXAM QUESTIONS FROM PREVIOUS YEARS. Curves. 1. Let s be the arclength parameter. γ (t) = (1, f ). ds = |γ (t)| = 1 + f 2 . dt dγ 1 = (1, f ) ds 1+f 2 d dγ 1 ff (1, f ) = (0, f ) − 2 dt ds (1 + f 2 )3/2 1+f 1 = (−f f , f ) (1 + f 2 )3/2 d2 γ f = (−f , 1) 2 ds (1 + f 2 )2 d2 γ f curvature = = 2 ds (1 + f 2 )3/2 Curvature of the parabola y = kx2 is 2k . (1 + 4k 2 x2 )3/2 2. Let t be the arclength parameter for α and let s be the arclength parameter for β (t) = Typeset by AMS-TEX 1 2 SOLUTIONS TO FINAL EXAM QUESTIONS FROM PREVIOUS YEARS. α (t). Let t, n, b be the Frenet frame for α. dβ d2 α = 2 = k n. dt dt ds dβ = = k. dt dt dβ = n. ds dn d dβ = = −k t − τ b. dt ds dt d2 β −k t − τ b τ = = −t − b. 2 ds k k 2 dβ τ2 curvature of β = = 1+ 2. ds2 k Tangent space and derivatives of maps. 1. There are misprints in this question. Let us assume that the cylinder is x2 + y 2 = 1 and that the tangent vector in T(1,0,0) is (0, 1, 1). I..

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