LINE INTEGRALS AND GREEN’S THEOREM 235 
I. 
//<- cos a + B cos /3 + C cos y) dS (outer normal) 
s 
//<■* cos a + B cos ¡3 + C cos y) cLS (inner normal). 
s 
The theorem embodied in this equation (in either form) is of 
fundamental importance, and we shall point out presently a number 
of its applications. It will be shown in Chap. XIII that each side 
of the equation is invariant of any rigid motion of the axes, or of 
any transformation to other Cartesian axes, provided merely that a 
right-handed system does not go over into a left-handed system. 
In the proofs given or indicated above it is tacitly assumed that 
the surface of V is cut by a parallel to the axis in question at most 
in two points or a single line-segment. It is sufficient for the needs 
of practice to restrict ourselves to such regions V as can be cut up 
into a finite number of regions V\, Vi, •••, for each of which this is 
true. On writing down equations (2) or I. for each of these regions 
and adding, the corresponding equation for V results. 
These theorems are known in the literature as Green's Theorem or 
Gauss’s Theorem; cf. § 4, end. 
EXERCISES 
1. Show that the integral 
jJ (x cos a -f y cos /3 -f- z cos y) dS, 
S 
where a, (3, y refer to the outer normal, is equal to three times the 
volume of the region. 
2. Setting A = u — , B — u~ 
show that 
du dv dudv du d\i 
dx dx dy dy dz dz 
’W 
7 
V V 
s 
where dv/dn is the directional derivative of v along the inner normal.