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Unformatted text preview: [Return HW; hand out time sheets] Prof. Tibor Beke will substitute for me on Thursday and Friday of this week. Another perspective on the “ v ′ ( t ) =  v ( t ) ′ for all t ” problem we discussed last time: What values can the left hand side take? What about the right hand side? Section 10.8: Arc length and curvature Three formulas for the curvature κ of a space curve: κ =  d T / ds  (the definition; requires the arclength parametrization) κ ( t ) =  T ′ ( t )/ r ′ ( t ) (any parametrization will do) κ ( t ) =  r ′ ( t ) × r ′′ ( t ) /  r ′ ( t ) 3 An intuitive description of κ =  T ′ ( t )/ r ′ ( t ): Since the length of T ( t ) is constant,  T ′ ( t ) is the rate of change of direction, or turning, of the unit tangent, and  r ′ ( t ) measures the speed along the curve. So κ is the rate of turning of the unit tangent divided by the speed along the curve. For a circle in the plane parametrized in the standard way, r ( t ) = 〈 a...
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 Fall '11
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 Velocity, Osculating circle, Prof. Tibor Beke

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