Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Q. 6 (page 889)

Let C be the graph of the vector-valued function r(t). Define the curvature at a point on C.

Short Answer

Expert verified

The curvature is defined as the rate of change of the tangent vector with respect to arc length.

Step by step solution

01

Step 1. Given information.

We have to define the curvature at a point on C where C be the graph of the vector-valued function r(t).

02

Step 2. Definition of the curvature

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=dTds

That is, the curvature is defined as the rate of change of the tangent vector with respect to arc length.

03

Step 3. Curvature k of c.

The Curvature k of c at a point on the curve is given by :

(a) k=T′(t)r′(t)where T(t) is the unit tangent vector to the r(t).

(b)k=r′(t)×r′′(t)r′(t)3

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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

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

Every description of the DNA molecule says that the strands of the helices run in opposite directions. This is meant as a statement about chemistry, not about the geometric shape of the double helix. Consider two helices

h1(t)=⟨cos⁡t,sin⁡t,αt⟩andh2(t)=⟨sin⁡t,cos⁡t,αt⟩

(a) Sketch these two helices in the same coordinate system, and show that they run geometrically in different directions.

(b) Explain why it is impossible for these two helices to fail to intersect, and hence why they could not form a configuration for DNA.

Completethefollowingdefinition:Ifr(t)=x(t),y(t),z(t)isadifferentiablepositionfunction,thenthevelocityvectorv(t)is....

Letr(t)=x(t),y(t),z(t).Provideadefinitionforthecontinuityofthevectorfunctionratapointcinthedomainofr.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free