220 
CALCULUS 
3. Line Integrals. The limits (3) and (5), § 2, are typical illus 
trations of liow line integrals come into mathematics. Let S be a 
region of the (x, y)-plane, and let 
<7: *=/(«)> t 0 St^t u 
be a regular curve lying in S. Let F(x, y, t) depend (continuously) 
on the point (x, y) in S and the point t of C. Divide C into n arcs 
by the points s 0 = 0, s 1? •••, s n _ 1} s n = l, where s denotes the arc, and 
l, the length of C. Let (x[, y' k ) be an arbitrary point of the ft-th arc, 
(«*_!, s k ), and let t" be any second point of the same arc. Then 
(!) lim 2 F ( x 'k, y'k, t") As* = f Fcls ■ 
n=x S J 
C 
is defined as the line integral of the function F along the curve C. 
That this limit exists is clear from Duhamel’s Theorem, since 
i 
(2) lim ^ F(x ki y k , t k ) As = / Fds 
n=x *=i J 
0 
is the ordinary integral of F{x, y, t), a continuous function of s. 
It is particularly to be remarked that, in the definition (1), As* 
is not a signed quantity, but is essentially positive. Thus the value 
of the line integral (1) does not depend on the sense of integration 
along C. We might equally well integrate in the opposite sense; 
the result would be the same. On the other hand, the line integrals 
presently to be defined are signed quantities. Reversal of the sense 
of integration along C reverses the signs of these integrals.* 
Definition of the Line Integral I Pdx 4- Qdy. 
C 
Let P be a function of (#, y), continuous throughout S. Let C be 
given as before, and let C be divided into n arcs by the points 
(x k , y k ). Form the sum : 
( 3 ) 2} > Vk) Ax k i Ax k = x k - x k _y, 
*=i 
* It might seem that the integral for the work, W = j* F cos f ds, is an excep- 
c 
tion, since reversing the sense in which the particle describes C reverses the sign 
of the work. But when the sense is reversed, f is replaced by its supplement, 
tt — i/-, and thus the sign of cos f is reversed.