   Chapter 13.3, Problem 25E

Chapter
Section
Textbook Problem

# Find the curvature of r(t) = ⟨t, t2, t3⟩ at the point (1, 1, 1).

To determine

To find: The curvature of r(t)=t,t2,t3 at point (1,1,1) .

Explanation

Given data:

The vector function r(t)=t,t2,t3 and point (1,1,1) .

Formula used:

Consider the three three-dimensional vectors such as a=a1,a2,a3 and b=b1,b2,b3 .

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(t) .

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 (|a|) .

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

Here,

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

The point on vector function r(t)=t,t2,t3 is (t,t2,t3) . The curvature is located at point (1,1,1) .

Hence equate the components of points (t,t2,t3) and (1,1,1) .

t=1

t2=1t=1t=±1

t3=1t=(1)13t=1

Therefore, the value of t is 1 from conclusions.

Write the vector function.

r(t)=t,t2,t3

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

r(t)=ddtt,t2,t3=ddt(t),ddt(t2),ddt(t3) {ddx(x)=1,ddx(xn)=nxn1}

r(t)=1,2t,3t2 (4)

Substitute 1 for t,

r(1)=1,2(1),3(1)2=1,2,3

Find the value of |r(1)| by using equation (3)

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