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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 arc-length 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
- Velocity, Osculating circle, Prof. Tibor Beke