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Chapter 11: Q. 9 (page 900)

Giveprecisemathematicaldefinitionsordescriptionsofeachoftheconceptsthatfollow.Thenillustratethedefinitionwithagraphoranalgebraicexample.theprincipalunitnormalvectoratr(t)

Short Answer

Expert verified

N(t)=T'(t)T'(t)

Step by step solution

01

Step 1. Given information is:

A differentiable vector function r(t)

02

Step 2. Principal Unit Normal Vector

Letr(t)beadifferentiablevectorfunctiononsomeintervalI鈯�鈩�suchthatthederivativeoftheunittangentvectorT'(t)鈮�0onI.Theprincipalunitnormalvectoratr(t)isdefinedtobe:N(t)=T'(t)T'(t).

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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).)

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

Using the definitions of the normal plane and rectifying plane in Exercises 20 and 21, respectively, find the equations of these planes at the specified points for the vector functions in Exercises 40鈥�42. Note: These are the same functions as in Exercises 35, 37, and 39.

r(t)=et,e鈭�t,2tatt=0

Statewhatitmeansforavectorfunctionr(t)=x(t),y(t),z(t)tobeintegrable.

If 伪, 尾, and 纬are nonzero constants, the graph of a vector function of the formrole="math" localid="1649577570077" rt=伪t,尾t2,纬t3 is called a twisted cubic. Prove that a twisted cubic intersects any plane in at most three points.
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