A5 - ~ 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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 222, Fall 2011 Calculus 3 Assignment 3, due Friday, October 21, 2011 1. For the helix defined parametrically by ~ r ( t ) = h 3cos t, 3sin t, 4 t i , (a) find the arc length s ( t ) travelled from 0 to t > 0, and specifically from 0 to 6 π ; (b) find the parametrization of the curve with respect to arc length from 0 to t > 0; (c) find the unit tangent ~ T ( t ) and find ~ T (6 π ); (d) find the principle unit normal ~ N ( t ) and ~ N (6 π ); (e) find the binormal vector ~ B ( t ) and ~ B (6 π ); (f) find the equation of the osculating plane at t = 6 π . (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 = 6 π . 2. For the curve defined parametrically by ~ r ( t ) = h 1 2 t 5 , 1 3 t 5 , 1 6 t 5 i , (a) find the arc length s ( t ) travelled from 0 to t > 0, and specifically from 0 to 1; (b) find the parametrization of the curve with respect to arc length from 0 to t > 0; (c) find the unit tangent ~ T ( t ) and find ~ T (1); (d) find the principle unit normal ~ N ( t ) and ~ N (1); (e) find the binormal vector
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 01/11/2012 for the course MATH 100 taught by Professor Loveys during the Fall '11 term at McGill.

Page1 / 2

A5 - ~ 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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online