   Chapter 13.3, Problem 65E

Chapter
Section
Textbook Problem

# Use the formula in Exercise 63(d) to find the torsion of the curve r ( t ) = 〈 t , 1 2 t 2 , 1 3 t 3 〉 .

To determine

To find: The torsion of the curve r(t)=t,12t2,13t3 .

Explanation

Formula:

Write the expression for torsion of curve r(t) .

τ=(r(t)×r(t))r(t)|r(t)×r(t)|2 (1)

Here,

r(t) is first derivative of r(t) ,

r(t) is second derivative of r(t) , and

r(t) is third derivative of r(t) ,

Consider the two three-dimensional vector functions such as u(t)=u1(t),u2(t),u3(t) and v(t)=v1(t),v2(t),v3(t) .

Cross product of vectors:

Write the expression for cross product of vectors u(t) and v(t) (u(t)×v(t)) .

u(t)×v(t)=|ijku1(t)u2(t)u3(t)v1(t)v2(t)v3(t)|=[(u2(t)v3(t)v2(t)u3(t))],[(u1(t)v3(t)v1(t)u3(t))],[(u1(t)v2(t)v1(t)u2(t))]

Dot product of vectors:

Write the expression for dot product of vectors u(t) and v(t) (u(t)v(t)) .

u(t)v(t)=u1(t),u2(t),u3(t)v1(t),v2(t),v3(t)=u1(t)v1(t)+u2(t)v2(t)+u3(t)v3(t)

Find the value of r(t) .

r(t)=ddtt,12t2,13t3=ddt(t),ddt(12t2),ddt(13t3)=1,12(2t),13(3t2) {ddx(xn)=nxn1}=1,t,t2

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

r(t)=ddt1,t,t2=ddt(1),ddt(t),ddt(t2)=0,1,2t {ddx(k)=0,ddx(x)=1,ddx(xn)=nxn1}

Apply differentiation with respect to t on both sides of equation

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