LINE INTEGRALS AND GREEN’S THEOREM 
Discuss the theorem of the text for the case of a region V which 
can be rendered linearly simply connected by the introduction of a 
finite number of diaphragms, like the cuts of the two-dimensional 
case. 
Theorem 2. Let P, Q, R be tivo functions which, together with 
the derivatives that enter below, are continuous throughout a linearly 
simply connected region V of space. In order that 
f 
Pdx-{- Q dy + R dz = 0, 
where C is any simple closed curve lying wholly within V, it is necessary 
and sufficient that 
dR 
dy 
cP 
dz 
f=o, 
dx 
8Q_ cP = o 
dx 
dy 
The proof of this theorem is also given in § 10. 
9. Green’s Theorem in Three Dimensions. Let P be a function 
of (x, y, z), continuous, together with dP/dz, within and on the 
boundary of a region V. Form the triple integral 
It can be evaluated by means of the iterated integral, Chap. IV, 
§ 2: 
rdP 
rdP 7 
— dz 
dz 
V s z 0 
= II P(x, y, ZOdS- f CP(x, y, Z„)dS, 
s ^ *s 
where S denotes the projection of V on the (x, ?/)-plane. 
These latter integrals can be expressed in terms of surface inte 
grals taken over the two nappes * of the boundary of V, 
* These nappes can be conveniently visualized as follows. Think of V as an 
opaque solid, and rays of light descending parallel to the axis of z. The part 
of the boundary illumined will be the upper nappe ; the dark part of the boun 
dary, the lower nappe. Moreover, S is the shadow cast by this solid on the 
{x, y)-plane.