LINE INTEGRALS AND GREEN’S THEOREM 237 
10. Stokes’s Theorem. Let P, Q, and R be three functions of 
(x, y, z) which, together with their first partial derivatives, are con 
tinuous throughout a region V of space, and let N be a surface lying 
in V and bounded by the curve C. Let a, /3, y be the direction 
angles of the normal to S, chosen in a suitable sense. Then Stokes’s 
Theorem asserts the truth of the equation: 
where the line integral is extended over C in a sense dependent on 
the choice of sense for the normal to S. 
The theorem is not true in general for unilateral surfaces (cf. infra), 
but it holds for all tico-sided surfaces. We begin by proving it for 
a restricted case, and are able then with ease to pass to the general 
case. 
A Restricted Case. Let S be given by the equation 
(1) Z = w(x,y), 
where w, together with its first derivatives, is continuous within 
and on the boundary of a region S' of the (x, y)-plane, and where 
S lies within the region V It is furthermore assumed that S' is 
the kind of region considered in § 4, to which Green’s Theorem is 
applicable. 
Consider the integral 
(2) I Pdx + Q dy H- Rdz, 
C 
taken in that sense along C which corresponds to the positive sense 
of description of the boundary T of S'. This integral can be ex 
pressed by a line integral over T as follows : 
(3) (P + R cdj) dx + ( Q + R <i> 2 ) dy. 
r 
For, the value of dz anywhere on S is 
dz = wxdx + <x) 2 dy,