Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

2.1 DIFFERENTIATION


Show that the radius of curvature at the point(x, y)on the ellipse
x^2
a^2

+


y^2
b^2

=1


has magnitude(a^4 y^2 +b^4 x^2 )^3 /^2 /(a^4 b^4 )and the opposite sign toy. Check the special case
b=a, for which the ellipse becomes a circle.

Differentiating the equation of the ellipse with respect toxgives


2 x
a^2

+


2 y
b^2

dy
dx

=0


and so
dy
dx


=−


b^2 x
a^2 y

.


A second differentiation, using (2.13), then yields


d^2 y
dx^2

=−


b^2
a^2

(


y−xy′
y^2

)


=−


b^4
a^2 y^3

(


y^2
b^2

+


x^2
a^2

)


=−


b^4
a^2 y^3

,


where we have used the fact that (x, y) lies on the ellipse. We note thatd^2 y/dx^2 , and hence
ρ, has the opposite sign toy^3 and hence toy. Substituting in (2.19) gives for the magnitude
of the radius of curvature


|ρ|=



∣∣



[


1+b^4 x^2 /(a^4 y^2 )

] 3 / 2


−b^4 /(a^2 y^3 )



∣∣



=


(a^4 y^2 +b^4 x^2 )^3 /^2
a^4 b^4

.


For the special caseb=a,|ρ|reduces toa−^2 (y^2 +x^2 )^3 /^2 and, sincex^2 +y^2 =a^2 ,thisin
turn gives|ρ|=a, as expected.


The discussion in this section has been confined to the behaviour of curves

that lie in one plane; examples of the application of curvature to the bending of


loaded beams and to particle orbits under the influence of a central forces can be


found in the exercises at the ends of later chapters. A more general treatment of


curvature in three dimensions is given in section 10.3, where a vector approach is


adopted.


2.1.10 Theorems of differentiation

Rolle’s theorem

Rolle’s theorem (figure 2.5) states that if a functionf(x) is continuous in the


rangea≤x≤c, is differentiable in the rangea<x<cand satisfiesf(a)=f(c)


then for at least one pointx=b,wherea<b<c,f′(b) = 0. Thus Rolle’s


theorem states that for a well-behaved (continuous and differentiable) function


that has the same value at two points either there is at least one stationary point


between those points or the function is a constant between them. The validity of


the theorem is immediately apparent from figure 2.5 and a full analytic proof will


not be given. The theorem is used in deriving the mean value theorem, which we


now discuss.

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