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Chapter 11: Q. 12 (page 900)

Giveprecisemathematicaldefinitionsordescriptionsofeachoftheconceptsthatfollow.Thenillustratethedefinitionwithagraphoranalgebraicexample.anarclengthparametrizationforthegraphofadifferentiablefunctionr(t)

Short Answer

Expert verified

鈭�cdr'(t)dt=d鈭�cforeverycandd鈭�I.

Step by step solution

01

Step 1. Given information is:

A differentiable vector function r(t) with graph C

02

Step 2. Arc Length Parametrization

LetCbethegraphofadifferentiablevectorfunctionr(t)definedonanintervalI.Thefunctionr(t)issaidtobeanarclengthparametrizationforCif鈭�cdr'(t)dt=d鈭�cforeverycandd鈭�I.Also,Cissaidtobeparametrizedbyarclength.

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Most popular questions from this chapter

Let k be a scalar and r(t)be a differentiable vector function. Prove that ddt(kr(t))=kr'(t). (This is Theorem 11.11 (a).)

Let r(t)=x(t),y(t),z(t),t鈭�[a,b], be a vector-valued function, where a < b are real numbers and the functions x(t), y(t), and z(t)are continuous. Explain why the graph of r is contained in some sphere centered at the origin.

For each of the vector-valued functions, find the unit tangent vector.

r(t2)=(t2+5,5t,4t3)

In Exercises 19鈥�21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

Domain

t鈭�鈩�,r2=1,1,r0=-3,3,r-2=-5,-5limt鈫�-鈭�x(t)=鈭�,limt鈫�-鈭�y(t)=-鈭�,andlimt鈫�鈭�r(t)=0,0.

Find parametric equations for each of the vector-valued functions in Exercises 26鈥�34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

r(t)=(cos2t,4int,t)fort鈭�[0,2蟺]
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