Transcription of 1 Reparametrization With Respect to Arc Length
1 Multivariate Calculus; Fall 2013S. JamshidiMATH 231: Calculus of Several VariablesSection 1, 107 Ag Sc & Ind Bldg,TR 9:05 AM - 9:55 AM1 Reparametrization With Respect to Arc LengthWe begin with our familiar formula for arc Length . Given a vector function~r (t), we can calculatethe Length fromt=atot=basL= ba|~r (t)|dtWe can actually turn this formula into a function of time. That is, we can create a functions(t) that measures how far we ve traveled from~r(a) at timet. We replacebwitht. To keep thingsclear, let s use the extra variable .s(t) = ta|~r ( )|d We can use this function to reparametrize~r(t) so that our input is the distance we traveled fromt= clarify, let s look at some ~r(t) = 2t,3 sin(2t),3 cos(2t) by its arc Length starting from(0,0,3). Use this information to determine the position after traveling , let s figure out for what value oftwill we get the point (0,0,3). We can do this by settingeach entry equal to the corresponding value. First, we consider,2t= 0 = t= 0 Ift= 0, then we need 3 sin(0) = 0 and 3cos(0) = 3, which are both true.
2 So, this happens att= is our initial , let s figure outs(t).First,~r (t) = 2,6 cos(2t), 6 sin(2t) 1 of 4 Multivariate Calculus; Fall 2013S. JamshidiWe now plug ins(t) = t0 4 + 36 cos2(2 ) + 36 sin2(2 )d = t0 4 + 36d = t0 40d =t 40 Once we have our expressions=t 40we can solve fort. When we do, we gett=s 40We then plug into~r(t) to get our final answer.~r(s) = 2s 40,3 sin(2s 40),3 cos(2s 40) = s 10,3 sin(s 10),3 cos(s 10) Finally, we plug in the distance 10 to determine the location.~r( 10) = 10 10,3 sin( 10 10),3 cos( 10 10) = ,0, 3 Example ~r(t) = sin(t),cos(t),1 by its arc Length starting from( 1,0,1).We re going to do the same steps as the previous problem. We begin with figuring out forwhat value oftwill we get the point ( 1,0,1). We can do this by setting each entry equal to thecorresponding value. First, we consider, sin(t) = 1 = t= 2 Ift= /2, then we need cos( /2) = 0, which is true. So, this is our initial of 4 Multivariate Calculus; Fall 2013S.
3 JamshidiNow, let s figure outs(t).Let s calculate the derivative.~r (t) = cos(t), sin(t),0 We now plug ins(t) = t /2 cos2( ) + sin2( )d = t /2 1d =t /2 Once we have our expressions=t /2we can solve fort. When we do, we gett=s+ /2We then plug into~r(t) to get our final answer.~r(s) = sin(s+ /2),cos(s+ /2),1 = cos(s),sin(s),1 Practice Problems1. Reparametrize~r(t) = 2t+ 1,4t,3t 1 by its arc Length starting from (3,4,2). Use this information to determine the position aftertraveling 5 Reparametrize~r(t) = sin(3t), cos(3t),4t by its arc Length starting from (0, 1,0). Use this information to determine the position aftertraveling Reparametrize~r(t) = sin(t2),cos(t2),t 2 by its arc Length starting from (0,1,0).3 of 4 Multivariate Calculus; Fall 2013S. Jamshidi4. Reparametrize~r(t) = rsin(t),rcos(t),0 by its arc Length starting from (0,r,0).4 of 4
