Calculus: Analytic Geometry and Calculus, with Vectors

420 Curves, lengths, and curvatures

being no reason for hope that the three numbers tk *, and t***are

the same. Thus

(7.252) JCJ = lim E [(tk )l2 + [y'(tk *)]2 + [z'(t,E**)]2 At,.

This looks much like a Riemann sum formed for the function f having

values

(7.253) f(t) = [x'(t)12 + [y'(0]2 + [z'(t)]2,

and there is no difficulty in the important special case in which x(t)

and z(t) = 0 for each t. Except in special cases, we must consider con-

sequences of the fact that tk, t)**, and tk*** may be different. Since

difficulties of this nature (and they were real difficulties in the old days)

are usually associated with the name of Duhamel (1797-1872), we can

call this a Duhamel difficulty. To treat all coordinates alike and to be

precise about this matter, we let Tk be the center of the interval tk_1 <

t < tk. We can then put (7.252) in the form

(7.254) ICI = lim j, [f(Tk) + 5A:] Mtk,

JPI-.0 k=1

where

(7.255) Sk = [x (tk )]2 + [y'(tk *)]2 + [Z'(tk**)]2

[x'(Tk)12 + [y'(Tk)]2 + [Z'(Tk)]2.

To prove the desired result

(7.26) JCJ = f bf(t) dt = f ba a [XI(t)12 + [y'(t)]2 + [z'(t)]2 dt,

it is therefore sufficient (and also necessary) to prove that

(7.261) lim Sk Qtk = 0.

IPI-.o k=1

It is easy to originate the correct idea that the numbers tk, tk**, and t***

are all near Tk, that continuity of the functions x', y', z' implies that the

two terms on the right side of (7.255) are nearly equal, that the numbers

Sk are all near 0, and hence that the sum in (7.261) is near 0, whenever the

norm JP1 of P is small. The easy way to make this precise is to use the

fact that the first term in the right member of (7.255) is a continuous

function of three variables. Using only functions of one variable, we

can let 0 < E < e and choose a number S > 0 such that the numbers

a, ,e, and y defined by

(7.262) [x'(u)]2 - [x'(a)]2 = a, [y'(u)]2 - [y'(v)]2 = fl,

[z'(u)]2 - [z'(v)]2 = 'Y