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Chapter 11: Q. 7 (page 889)

What makes Definition 11.21 for curvature easy to understand? What makes it difficult to use?

Short Answer

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The reason has been explained.

Step by step solution

01

Step 1. Given information.

We have to explain What makes Definition 11.21 for curvature easy to understand and What makes it difficult to use.

02

Step 2. Definition

The definition of curvature is, let c be the graph of a vector function r(s) defined on an interval I, parametrized by arc length s, and with unit tangent vector T.

The curvature k of c at a point on the curve is the scalar given by k=dTdsi.e. the rate of change of the tangent vector with respect to arc length is the curvature.

03

Step 3. Explanation

Let a helix is defined by r(t)=鉄�sin鈦�伪t,cos鈦�伪t,尾t鉄�on[a,b]where伪,尾,aandb

The derivative is r鈥�(t)=鉄�伪cos鈦�伪t,鈭�伪sin鈦�伪t,尾鉄�

The arc length is :

鈭�ab鈥�r鈥�(t)dt=鈭�ab鈥�伪2cos2鈦�伪t+伪2sin2鈦�伪t+尾2dt=鈭�ab鈥�伪2+尾2dt=伪2+尾2tab=伪2+尾2(b鈭�a)

Thus, the arc length of r(t) is(b鈭�a)伪2+尾2

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Most popular questions from this chapter

Let r(t)=xt,yt,t鈭�[a,鈭�), be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in a circle centered at the origin. (Hint: Graph the functions r1t=1t,1tand r2t=t,t,both with domain 摆1,鈭�).)

Given a vector-valued function r(t) with domain 鈩�, what is the relationship between the graph of r(t) and the graph of kr(t), where k > 1 is a scalar?

Let r1(f)and r2(t) be differentiable vector functions with three components each. Prove that

ddtr1(t)-r2(t)=r1'(t)-r2(t)+r1(t)-r2'(t)鈭�(This is Theorem 11.11 (c).)

Letr(t)=x(t),y(t),z(t).Whatisthedefinitionoflimt鈫�cr(t)?

For each of the vector-valued functions in Exercises, find the unit tangent vector and the principal unit normal vector at the specified value of t.

rt=sin2t,cos2t,t,t=蟺4
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