Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.1 DIFFERENTIATION

C

P

Q

ρ

θ θ+∆θ

∆θ

x

f(x)

Figure 2.4 Two neighbouring tangents to the curvef(x) whose slopes differ

by ∆θ. The angular separation of the corresponding radii of the circle of

curvature is also ∆θ.

pointPon the curvef=f(x), with tanθ=df/dxevaluated atP. Now consider

also the tangent at a neighbouring pointQon the curve, and suppose that it

makes an angleθ+∆θwith thex-axis, as illustrated in figure 2.4.

It follows that the corresponding normals atPandQ, which are perpendicular

to the respective tangents, also intersect at an angle ∆θ. Furthermore, their point

of intersection,Cin the figure, will be the position of the centre of a circle that

approximates the arcPQ, at least to the extent of having the same tangents at

the extremities of the arc. This circle is called thecircle of curvature.

For a finite arcPQ, the lengths ofCPandCQwill not, in general, be equal,

as they would be iff=f(x)werein fact the equation of a circle. But, asQ

is allowed to tend toP,i.e.as∆θ→0, they do become equal, their common

value beingρ, the radius of the circle, known as theradius of curvature. It follows

immediately that the curve and the circle of curvature have a common tangent

atPand lie on the same side of it. The reciprocal of the radius of curvature,ρ−^1 ,

defines thecurvatureof the functionf(x) at the pointP.

The radius of curvature can be defined more mathematically as follows. The

length ∆sof arcPQis approximately equal toρ∆θand, in the limit ∆θ→0, this

relationship definesρas

ρ= lim

∆θ→ 0

∆s

∆θ

=

ds

dθ

. (2.15)

It should be noted that, assincreases,θmay increase or decrease according to

whether the curve is locally concave upwards (i.e. shaped as if it were near a

minimum inf(x)) or concave downwards. This is reflected in the sign ofρ,which

therefore also indicates the position of the curve (and of the circle of curvature)