   Chapter 10.2, Problem 69E

Chapter
Section
Textbook Problem

The curvature at a point P of a curve is defined as κ = | d ϕ d s | where φ is the angle of inclination of the tangent line at P, as shown in the figure. Thus the curvature is the absolute value of the rate of change of φ with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at P and will be studied in greater detail in Chapter 13. (a) For a parametric curve x = x(t), y = y(t), derive the formula κ = | x ˙ y ¨ − x ¨ y ˙ | [ x ˙ 2 + y ˙ 2 ] 3 / 2 (b) where the dots indicate derivatives with respect to t, so x ˙ = d x / d t . [Hint: Use φ = tan−1(dy/dx) and Formula 2 to find dφ/dt. Then use the Chain Rule to find dφ/ds.] (c) By regarding a curve y = f(x) as the parametric curve x = x, y = f(x), with parameter x, show that the formula in part (a) becomes κ = | d 2 y / d x 2 | [ 1 + ( d y / d x ) 2 ] 3 / 2 (a)

To determine

To derive: The formula k=|x˙y¨y˙x¨|(x˙2+y˙2)3/2 for the parametric equation for the curve x=x(t), y=y(t).

Explanation

Given:

Write the parametric equation of the curve.

The parametric equation for the variable x is as below.

x=x(t)

The parametric equation for the variable y is as below.

y=x(t)

From the above figure (1), the curves curvature points at P is defined as k=dϕds.

At the tangent line at P, ϕ is the angle of inclination and it is given by ϕ=tan1(dydx).

Calculation:

The variable ϕ is the angle of inclination, ϕ=tan1(dydx).

dϕdt=ddttan1(dydx)

dϕdt=11+(dydx)2ddt(dydx) (1)

Use the chain rule.

Write the chain rule for expression dydx.

dydx=dydtdxdt=y˙x˙

Substitute (y˙x˙) for (dydtdxdt) in equation (1).

dϕdt=11+(y˙x˙)2ddt(y˙x˙)=x˙2(x˙2+y˙2)×x˙y¨y˙x¨x˙2

Applying quotient rule for y¨=d2ydt2 and x˙=d2xdt2

(b)

To determine

To derive: The formula k=|d2ydx2|[1+(dydx)2]3/2 for the parametric equation for the curve x=x,y=f(x) with parameter x.

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