236 
CALCULUS 
3. Setting 
show that 
Aw — — 4- — 4- 
dx 2 €y l dz 2 ’ 
III (u Ay — y Aw) dV = — II 
dv du\ ,c 
u z v Q- dS > 
dn on) 
where n refers to the inner normal. This equation is properly 
known as Green’s Theorem ; cf. § 4. 
4. Prove that 
III 
d 2 u , d 2 u d-u 
dx 1 cy 2 dz 2 
dV-- 
-II 
dS. 
5. If u is a solution of Laplace’s equation, 
chi dhi dhi _ q 
dx 2 dy 2 dz 2 ’ 
show that 
II 
^dS = 0. 
dii 
6. If u is a solution of Laplace’s equation, show that 
//./IS 
+ 
du\ 2 fdu\ 2 ' 
cy ■ 
cz 
dV- 
-II 
u^-dS. 
dn 
7. If it is a solution of Laplace’s equation, show that 
II 
u~dS<0. 
dn 
8. Let A, B, C be three functions which, together with the deriva 
tives that enter below, are continuous throughout the interior of a 
region V' of space. Let V be any region contained within V'; let 
S refer to the boundary of V, and let a, /3, y be the direction angles 
of the inner normal of S. In order that 
II 
(A cos a + B cos /3 + C cos y) dS = 0, 
it is necessary and sufficient that 
dA dB _,dC = 0 
dx dy dz 
Prove this proposition.