   Chapter 13.3, Problem 21E

Chapter
Section
Textbook Problem

# Use Theorem 10 to find the curvature.21. r(t) = t3 j + t2 k

To determine

To find: The curvature of r(t)=t3j+t2k using Theorem 10.

Explanation

Given data:

The vector function r(t)=t3j+t2k .

Formula used:

Consider the three three-dimensional vectors such as a=a1i+a2j+a3k and b=b1i+b2j+b3k .

Cross product of vectors:

Write the expression for cross product of vectors a and b (a×b) .

a×b=|ijka1a2a3b1b2b3|

a×b=(a2b3b2a3)i(a1b3b1a3)j+(a1b2b1a2)k (1)

Write the expression for curvature of vector function r according to theorem 10.

k(t)=|r(t)×r(t)||r(t)|3 (2)

Here,

r(t) is first derivative of vector function r(t) , and

r(t) is second derivative of vector function r(t) .

Write the expression for magnitude of vector a=a1i+a2j+a3k (|a|) .

|a|=a12+a22+a32 (3)

Here,

a1 , a2 and a3 are the x, y, and z-coordinates of vector respectively.

Write the vector equation.

r(t)=t3j+t2k

Apply differentiation with respect to t on both sides of equation.

r(t)=ddt(t3j+t2k)=ddt(t3)j+ddt(t2)k

r(t)=3t2j+2tk {ddx(xn)=nxn1} (4)

Find the value of |r(t)| by using equation (3).

|r(t)|=02+(3t2)2+(2t)2=9t4+4t2

Apply differentiation with respect to t on both sides of equation (4)

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Simplify the expressions in Exercises 97106. x1/2y2x1/2y

Finite Mathematics and Applied Calculus (MindTap Course List)

#### In Exercises 7-22, evaluate the expression. 22. 62.561.961.4

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### In problems 23-58, perform the indicated operations and simplify. 44.

Mathematical Applications for the Management, Life, and Social Sciences

#### If limx0sinxx=1, the angle x must be measured in: a) radians b) degrees c) it does not matter

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

#### The graph at the right has equation:

Study Guide for Stewart's Multivariable Calculus, 8th 