Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

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

=

ds
dx

dx

. (2.17)


Differentiating both sides of tanθ=df/dxwith respect toxgives


sec^2 θ


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

=

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.

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