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09.28 - [Return HW hand out time sheets Prof Tibor Beke...

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[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.

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For a circle in the plane parametrized in the standard way, r ( t ) = a cos t , a sin t (with a > 0), the angle of the unit tangent will change by 2 π as we go around the circle (a total of 2 π
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09.28 - [Return HW hand out time sheets Prof Tibor Beke...

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