234 
CALCULUS 
the upper nappe z = = fa (x, y) ; 
the lower nappe 2 0 , z = Z 0 = <£ 0 (æ, 2/). 
2/, 
/ 
o 
2 
FlG - 56 Thus the difference of the two double 
integrals is seen to have the value of the 
surface integral of P cos y taken over the entire surface 2, and so we 
have the result: 
(1) 
V 
2 
Similar formulas could have been obtained if we had started with 
partial derivatives with respect to x or y, the region V being now 
projected on the (y, z)-plane or the (x, z)-plane. The results in all 
three cases can be collected as follows : 
(2) 
V 
s 
V 
s 
where we have replaced the letter 2, as referring to the bounding 
surface, by the letter S. 
If we had used the inner normal, the sign of each right-hand side 
would have been reversed. 
On adding the three equations (2) together, we get: