Curves Suppose now that the curve C:r(t)=x(t)i+y(t)j+z(t)k is twice differentiable and r'(t)is never zero.Then at each point P(x(t),v(t),z())of the curve,there is a unit tangent vector: r'(t) (13.3.3) T(t)= lr')I Since lr'(t)>0,T(t)points in the direction of r(t)that is,in the direction of increas- ing t.Since T(t)l=1,we have T(t).T(t)=1.Differentiation gives T(t).T'(t)+T'(t).T(t)=0. Unit tangent vector (13.3.3),p.781 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Curves Unit tangent vector (13.3.3), p. 781
Curves If T'(t)0,then we can form what is called the principal normal vector: T'(t) (13.3.4) N(t)= T')川 This is the unit vector in the direction of T'(t).The normal line at P is the line through P parallel to the principal normal. Principal normal vector(13.3.4),p.781 Main Menu 00 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Curves Principal normal vector (13.3.4), p. 781
Curves The plane determined by T and N is called the osculating plane. Osculating plane,p.781 Main Menu
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Curves Osculating plane, p. 781 The plane determined by T and N is called the osculating plane
Example 4 In Figure 13.3.12 you can see a curve spiraling up a circular cylin- der.The curve is called a circular helix if the rate of climb is constant.The simplest parametrization for a circular helix takes the form r(t)=acos/i+asin!j+br k with a>0,b>0. circular helix Figure 13.3.12 Main Menu
Main Menu
Arc Length a t=b X Arc length (13.4.1),p.787,Figures 13.4.1-2 Main Menu
Main Menu Arc Length Arc length (13.4.1), p. 787, Figures 13.4.1-2
DEFINITION 13.4.1 ARC LENGTH L(C)= the least upper bound of the set of all lengths of polygonal paths inscribed in C. Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved
Arc Length THEOREM 13.4.2 ARC LENGTH FORMULA Let C be the path traced out by a continuously differentiable vector function r=r(t), t∈「a.b1. The length of C is given by the formula L(C)= r'(t川d. Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved. Arc Length
Example 1 Find the length of the curve r(r)=212i+4j from =0to=1. Main Meny C00e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved
Example 2 Find the length of the curve r(t)2 costi+2sintj+2k from1=0to1=π/2 and compare it to the straight-line distance between the endpoints of the curve. 02的 2.0.0 Main Meny c8S的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2003 © John Wiley & Sons, Inc. All rights reserved