Calculus
6th Edition
ISBN: 9781465208880
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 10.4, Problem 57PS
To determine
To calculate: the Frenet- Serret formula is to be proved.
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Show that for a plane curve the torsion T = 0.
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b. The vector function r(t)=sin 2ti-cos 2tj+t√5k determines a curve C'in space.
i. Find the unit tangent vector I and the principal unit normal N
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Chapter 10 Solutions
Calculus
Ch. 10.1 - Prob. 1PSCh. 10.1 - Prob. 2PSCh. 10.1 - Prob. 3PSCh. 10.1 - Prob. 4PSCh. 10.1 - Prob. 5PSCh. 10.1 - Prob. 6PSCh. 10.1 - Prob. 7PSCh. 10.1 - Prob. 8PSCh. 10.1 - Prob. 9PSCh. 10.1 - Prob. 10PS
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- Consider the helix r(t) A. The unit tangent vector T = = ( B. The unit normal vector N= ( D. The curvature x = (cos(1t), sin(1t), −1t). Compute, at t = 4: C. The unit binormal vector B = (arrow_forwardConsider the helix r(t) = (cos(-4t), sin(-4t), 3t). Compute, at t = 픔 A. The unit tangent vector T = ( 000 B. The unit normal vector N = (00 C. The unit binormal vector B=( 000 D. The curvature k = E|Oarrow_forwardUse the theorem given below to find the curvature of r(t) = √6t² i + 2tj + 2t³ k. Theorem: The curvature of the curve given by the vector function r is k(t) |r'(t) x r"(t)| \r(t)|3arrow_forward
- A curve C in the plane is defined by the parametric equations: x=t² i. Find the length of C from t=0 to 1-2 ii. Find the curvature of C at t = 1 b. The vector function r(t)=sin 2ti-cos2tj+t√√5k determines a curve C in space. i. Find the unit tangent vector T and the principal unit normal N ii. Determine the curvature of C at time t x=f² +4y=2²=1 iii. Determine the tangential and normal component of the acceleration vector. 5. Let f(x,y)=xln(x/y) + xy² a. Calculate f, and f b. Determine the directional derivative off at the point (2, 2) in the direction of the vector a=i-2j af Ət c. Suppose that x = ste' and y=2se. Calculate d. Determine an equation for the tangent plane to the surface == f(x, y) at the point (2, 2, 8) on the surface. 6. Let F(x, y, z)=2xy² +2y=²+2x²z. a. Determine the maximum directional derivative of F at the point (1, -1, 1). b. Find the directional derivative off at the point (-2, 1, -1) in the direction parallel to the line x=34t, y=2-t, z=3t. c. Determine…arrow_forwardProve that the curve r(t) = (a + bt" ,c + dt",e + ft), where a, b, c, d, e, and f are real numbers and p is a positive integer, has zero curvature. Give an explanation What must be shown to prove that r(t) has zero curvature? OA. It must be shown that the velocity, v, and the acceleration, a, are constant. GB. It must be shown that the magnitude of the cross product, axv, is zero, or that the unit tangent vector, T, is constant. OC. It must be shown that the cross product a xv is constant. O D. It must be shown that the dot product, a v, is zero, or that the acceleration, a, is constant. In this problem, show that the magnitude of the cross product, axv, is zero. To do so, first find the velocity, v. {a + bt,c+ dtP,e + ftP}arrow_forward
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