25 27] DEFINITE INTEGRALS. 45 



In like manner 



~ f 



dwj< 



If now u, v, w, are all functions of a variable , we have for the 

 derivative of the definite integral according to t, 



dF d f ., dFdu dF dv dF dw 



c?w f 



^ 



27. Double and Multiple Integrals. Suppose we consider 

 a continuous function of two variables, x varying from a to 6, and 

 y varying from g to h. We may represent f(x, y) geometrically as 

 the third coordinate z of a surface, erected perpendicular to the 

 plane of xy. If now we subdivide the interval ab by points 



a<x l < # 2 < ...... x s < # n _j< b, 



and the interval gh by points 



and draw through these points lines parallel to the axes of x and y, 

 dividing the plane into rectangles, and at a point in each rectangle 

 erect perpendiculars meeting the surface, the sum 



s=n r=m 



2 2 (x s - x s ^) (y r - y,- 



s=l r=l 



represents the volume of the rectangular prisms constructed on 

 the rectangles with sides x s x s _ l} y r y r ^ as bases, and altitudes 



If as we make the number of points of subdivision increase 

 without limit, the sum approaches a limit, this limit defines the 

 definite integral 



h s=n r=m 



> y) dx dy = lim lim 2 2 (x s - x^} (y r - y r -i)f s , r> 



g n=oo w=oo5=l r=l 



We shall find by reasoning similar to that used in 22 that 

 the condition for the existence of a limit is that the sum 



s=n r=m 



2 2 (sc 8 -x 8 , l )(y 8 -y s . l )D 8r) 



s=l r=l 



rb r 



I 



J a J 



