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Unformatted text preview: ~ B ( t ) and ~ B (1); (f) ﬁnd the equation of the osculating plane at t = 1. (g) Regarding ~ r ( t ) as giving the motion of a particle in space (with respect to time t ) ﬁnd the acceleration ~a ( t ) and the tangent and normal component of acceleration. Evaluate these at t = 1. 3. Suppose that ~ r ( t ) deﬁnes a curve parametrically for a ≤ t ≤ b , and that ~ r ( t ) 6 = ~ 0 on [ a,b ]. Show that d dt  ~ r ( t )  = ~ r ( t ) · ~ r 00 ( t )  ~ r ( t )  . [Hint: What is  ~ r ( t )  2 and what is its derivative?] 1 4. Suppose that ~ r ( s ) deﬁnes a curve parametrically with respect to arc length and that ~ r ( s ) is nonzero on the curve. Show that d ~ B ds is orthogonal to both ~ B ( s ) and ~ T ( s ). Conclude that there is a scalar function τ ( s ) such that d ~ B ds =τ ( s ) ~ N . (This function τ is known as the torsion of the curve.) 2...
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This note was uploaded on 01/11/2012 for the course MATH 100 taught by Professor Loveys during the Fall '11 term at McGill.
 Fall '11
 Loveys
 Arc Length

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