27, 28] DEFINITE INTEGRALS. 47 



If instead of a rectangle in the XF-plane we should take any closed 

 curve, given by an equation <f> (x, y) = 0, we could in like manner 

 divide its area into rectangles, erecting perpendiculars in each, 

 and define the definite integral as the limit of a similar sum for 

 the new field of integration. 



More generally, let f(M) be a function of a point M, moving 

 either in a plane or in space. If we divide any area S in the 

 plane or any volume r in space up into a number of parts, take the 

 value of/(.M)at any point within each of those parts, multiply 

 each value by the area or volume of the part in which it is taken, 

 and add together for all the parts into which the area or volume is 

 divided, the limit approached by this sum as the number of parts 

 increases without limit in such a way that each dimension of 

 every part approaches zero, if such a limit exists, is called the 

 definite integral of f(M) through the region in question. We 

 may write the integrals 



fjf(M)dS or jfjf(M)dr, 



respectively. In each case, the field of integration must be ex- 

 pressly specified. It may be easily shown that this definition is 

 equivalent to the preceding. 



A particular mode of subdivision is by drawing level lines 

 or surfaces for two or three orthogonal coordinates q lt q 2 , q 3 . We 

 have then, ( 20), 



7 _ rfgi dq 3 , dq,. dq 2 dq 3 



U>O - O-T = - ; - - ; - . 



Suppose that in two different sets of coordinates 

 q^qz, and p ly p a ,p a with parameters, h 1} h z ,h 3) and 



when taken through any equal finite portions of a volume r. 



Then when we consider the meaning of the definite integral 

 and its independence of the manner of subdivision, we see as in 

 ( 23) that the above integrals, being respectively equal to 



JJJ h l h z h 3 f(q l) q 2> q s ) dr and Jj J g^g z $ (p l} p 2 , p 3 ) dr, 



