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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 re-spect 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
- Arc Length