Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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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

. (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)

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