224 
CALCULUS 
The identities (A), (B), (C) are usually referred to in the litera 
ture as Green’s Theorem (1828) or Gauss’s Theorem (1813). Such 
identities go back, however, much further, appearing (in the case of 
volume integrals) as early as 1760/61 in work of Lagrange’s. The 
theorem of Ex. 5 below may, however, properly be called Green’s 
Theorem. Cf. a note in the author’s Funktionentheorie, Yol. I, 2d 
ed., p. 600. 
EXERCISES 
1. Extend the integral 
y dx — xdy 
x 2 + y 2 
in the positive sense over the boundary (i) of a circle whose centre 
is at the origin; (u) of a circular ring with its centre at the origin. 
Ans. (i) 2it', (ii) 0. 
2. Show that the integral ,— 
/ 
xdy — y dx, 
extended in the positive sense over the complete boundary of any 
region, is equal to twice the area of the region. 
3. Setting 
D dv dv 
F = u — , Q = u —, 
dx cy 
show that 
f s f^f) dS= f udv - 
4. Setting 
ZD 
show that 
//■(S'- 
s 
+ C f(f d S + d ^)as = - fu^ds. 
d Vj J J \dx dx dy dy) J dn 
s c 
5. Setting 
a chi , d 2 u 
^ u ~w + df' 
show that 
Jf(uAv-vAu)dS~ - j (»g-»|;)<&. 
s c 
This equation is properly known as Green’s Theorem.