Leibniz's Formula for Radius of Curvature

The radius r of a circle drawn through three infinitely nearby points on a curve in the (x, y)-plane satisfies

1/r = -d/dx (dy/ds)

where s denotes the arclength.  For example, if y = f[x], so ds = (1 + (f^′ [x])^2)^(1/2) dx, then

1/r = -d/dx (f^′ [x]/(1 + (f^′[x])^2)^(1/2)) = -f^′′[x]/(1 + f^′[x]^2)^(3/2)

If  the curve is given parametrically, y = y[t] and x = x[t], so ds = (x^′[t]^2 + y^′[t]^2)^(1/2) dt, then

1/r = -d(dy/ds)/dx = (y^′[t] x^′′[t] - x^′[t] y^′′[t])/(x^′[t]^2 + y^′[t]^2)^(3/2)

Changes

Consider three points on a curve  with equal distances Δs between the points.  Let α_I and α_II denote the angles between the horizontal and the segments connecting the points as shown.  We have the relation between the changes in y and α:

Sin[α] = Δy/Δs (2)

The difference between these angles, Δα, is shown near p_III.  

[Graphics:../HTMLFiles/Lect1_220.gif]

The angle between the perpendicular bisectors of the connecting segments is also Δα, because they meet the connecting segments at right angles.

These bisectors meet at the center of a circle through the three points on the curve whose radius we denote r.  The small triangle with hypotenuse r gives

Sin[Δα/2] = (Δs/2)/r (3)

Small Changes

Now we apply these relations when the distance between the successive points is an infinitesimal δs. The change

-δ Sin[α] = -δ(δy/δs) = Sin[α] - Sin[α - δα] = Cos[α]  δα + ϑ  δα, (4)

with ϑ≈0, by smoothness of sine (see above).  Smoothness of sine also gives,

Sin[δα/2] = δα/2 + η  δα, with η≈0

Combining this with formula (1.1.3) for the infinitesimal case (assuming r≠0), we get

δα = δs/r + ι  δα, with ι≈0

Now substitute this in (1.1.4) to obtain

-δ(δy/δs) = Cos[α] δs/r + ζ  δs, with ζ≈0

By trigonometry, Cos[α] = δx/δs, so

-δ(δy/δs)/δx = 1/r + ζ  δs/δx≈1/r, as long as δs/δx is not infinitely large.

Keisler's Function Extension Axiom allows us to apply formulas (1.1.3) and (1.1.4) when the change is infinitesimal, as we shall see.  We still have a gap to fill in order to know that we may replace infinitesimal differences with differentials (or derivatives), especially because we have a difference of a quotient of differences.  

First differences and derivatives have a fairly simple rigorous version in Robinson's theory, just using the differential approximation (1.1.1).  This can be used to derive many classical differential equations like the tractrix,  catenary, and isochrone, see: Chapter 5 Differenital Equations from Increment Geometry in Projects for Calculus: The Language of Change on my website at http://www.math.uiowa.edu/%7Estroyan/ProjectsCD/estroyan/indexok.htm

Second differences and second derivatives have a complicated history.  See

H. J. M. Bos, Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus, Archive for History of Exact Sciences, vol. 14, nr. 1, 1974.

This is a very interesting paper that begins with a course in calculus as Leibniz might have presented it.

The natural exponential

The natural exponential function satisfies

y[0] = 1
dy/dx = y

We can use (1.1.1) to find an approximate solution,

y[δx] = y[0] + y '[0]  δx = 1 + δx

Recursively,

y[2 δx] = y[δx] + y '[δx]  δx = y[δx]  (1 + δx) = (1 + δx)^2

y[3 δx] = y[2 δx] + y '[2 δx]  δx = y[2 δx]  (1 + δx) = (1 + δx)^3

:

y[x] = (1 + δx)^(x/δx), for x = 0, δx, 2 δx, 3 δx, ⋯

This is the product expansion ≈ (1 + δx)^(1/δx), for δx≈0.

No introduction to calculus is complete without mention of this sort of "infinite algebra" as championed by Euler as in

L. Euler, Introductio in Analysin Infinitorum, Tomus Primus, Lausanne, 1748. Reprinted as L. Euler, Opera Omnia, ser. 1, vol. 8. Translated from the Latin by J. D. Blanton, Introduction to Analysis of the Infinite, Book I, Springer–Verlag, New York, 1988.

A wonderful modern interpretaion of these sorts of computations is in

Mark McKinzie and Curtis Tuckey, Higher Trigonometry, Hyperreal Numbers and Euler's Analysis of Infinities, Math Magazine, vol. 74, nr. 5, Dec. 2001, p. 339-368

W. A. J. Luxemburg's reformulation of the proof of one of Euler's central formulas

Sin[z] = z Underoverscript[∏, k = 1, arg3] (1 - (z/(k π))^2)

appears in our monograph, Introduction to the Theory of Infinitesimals, Academic Press Series on Pure and Applied Math. vol 72, 1976, Academic Press, New York.


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