Mathematical Methods for Physics and Engineering - Matematica.NET

2.1 DIFFERENTIATION◮Show that the radius of curvature at the point (x, y) on the ellipsex 2a + y22 b =1 2has magnitude (a 4 y 2 + b 4 x 2 ) 3/2 /(a 4 b 4 ) and the opposite sign to y. Check the special caseb = a, for which the ellipse becomes a circle.Differentiating the equation of the ellipse with respect to x gives2xa + 2y dy2 b 2 dx =0and sodydx = − b2 xa 2 y .A second differentiation, using (2.13), then yields( ) ( )d 2 ydx = − b2 y − xy′= − b4 y22 a 2 y 2 a 2 y 3 b + x2= − b42 a 2 a 2 y , 3where we have used the fact that (x, y) lies on the ellipse. We note that d 2 y/dx 2 , and henceρ, has the opposite sign to y 3 and hence to y. Substituting in (2.19) gives for the magnitudeof the radius of curvature[ 1+b 4 x 2 /(a 4 y 2 ) ] 3/2|ρ| =∣ −b 4 /(a 2 y 3 ) ∣ = (a4 y 2 + b 4 x 2 ) 3/2.a 4 b 4For the special case b = a, |ρ| reduces to a −2 (y 2 + x 2 ) 3/2 and, since x 2 + y 2 = a 2 ,thisinturn gives |ρ| = a, as expected. ◭The discussion in this section has been confined to the behaviour of curvesthat lie in one plane; examples of the application of curvature to the bending ofloaded beams and to particle orbits under the influence of a central forces can befound in the exercises at the ends of later chapters. A more general treatment ofcurvature in three dimensions is given in section 10.3, where a vector approach isadopted.2.1.10 Theorems of differentiationRolle’s theoremRolle’s theorem (figure 2.5) states that if a function f(x) is continuous in therange a ≤ x ≤ c, is differentiable in the range a