Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

relative to the common tangent, above or below. Thus a negative value ofρ

indicates that the curve is locally concave downwards and that the tangent lies

above the curve.

We next obtain an expression forρ, not in terms ofsandθbut in terms

ofxandf(x). The expression, though somewhat cumbersome, follows from the

defining equation (2.15), the defining property ofθthat tanθ=df/dx≡f′and

the fact that the rate of change of arc length withxis given by

ds

dx

=

[

1+

(

df

dx

) 2 ]^1 /^2

. (2.16)

This last result, simply quoted here, is proved more formally in subsection 2.2.13.

From the chain rule (2.11) it follows that

ρ=

ds

dθ

=

ds

dx

dx

dθ

. (2.17)

Differentiating both sides of tanθ=df/dxwith respect toxgives

sec^2 θ

dθ

dx

=

d^2 f

dx^2

≡f′′,

from which, using sec^2 θ=1+tan^2 θ=1+(f′)^2 , we can obtaindx/dθas

dx

dθ

=

1+tan^2 θ

f′′

=

1+(f′)^2

f′′

. (2.18)

Substituting (2.16) and (2.18) into (2.17) then yields the final expression forρ,

ρ=

[

1+(f′)^2

] 3 / 2

f′′

. (2.19)

It should be noted that the quantity in brackets is always positive and that its

three-halves root is also taken as positive. The sign ofρis thus solely determined

by that ofd^2 f/dx^2 , in line with our previous discussion relating the sign to

whether the curve is concave or convex upwards. If, as happens at a point of

inflection,d^2 f/dx^2 is zero thenρis formally infinite and the curvature off(x)is

zero. Asd^2 f/dx^2 changes sign on passing through zero, both the local tangent

and the circle of curvature change from their initial positions to the opposite side

of the curve.