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Math
CalculusMultivariable Calculus8th Edition

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Chapter 13.3, Problem 21E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 38E
Problem 39E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 53E
Problem 55E
Problem 56E
Problem 58E
Problem 59E
Problem 60E
Problem 62E
Problem 63E
Problem 64E
Problem 65E
Problem 66E
Problem 67E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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=(a2b3−b2a3)i−(a1b3−b1a3)j+(a1b2−b1a2)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)=nxn−1} (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)

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Chapter 13.3,
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