MATHEMATICS 317 T2 - cos 1 = a r 3(a Prove that the torsion 2 2 2 2 3 3 2 2 ds d ds d ds d ds d ds d s r r r r r ⋅ × ⋅ = τ for a curve r s

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MATHEMATICS 317 Assignment #2 due at the beginning of class on Wednesday, January 20 th 1 (a) For a curve defined by , 0 ) , ( = y x f show that the radius of curvature R at any point of the curve is given by the expression . 2 ] [ 2 2 2 / 3 2 2 yy x xx y xy y x y x f f f f f f f f f R - - + = (b) Find the radius of curvature at the point ) 0 , ( a for the ellipse . 1 2 2 2 2 = + b y a x 2. (a) If r and θ are polar coordinates, show that the radius of curvature R at any point of the curve defined by ) ( θ f r = is given by the expression . 2 ] [ 2 2 2 / 3 2 2 f f f f f f R - + + = (b) Find the radius of curvature at any point lying on the cardioid
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Unformatted text preview: ). cos 1 ( + = a r 3. (a) Prove that the torsion 2 2 2 2 3 3 2 2 ) ( ds d ds d ds d ds d ds d s r r r r r ⋅ × ⋅ = τ for a curve r ( s ) defined in terms of its intrinsic arc length parameter s . (b) Use the formula derived in (a) to find the torsion at any point of the circular helix defined by . sin cos ) ( k j i r bt t a t a t + + =...
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This note was uploaded on 04/01/2010 for the course MATH 317 MATH 317 taught by Professor Bluman during the Spring '10 term at The University of British Columbia.

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