LINE INTEGRALS AND GREEN’S THEOREM 239 
This integral has the same value as the surface integral, taken 
over S: 
But this is precisely the integral that stands on the left of 
Equation I., and hence the theorem is proved for this case. It is 
to be observed that the positive sense of C, the inner normal to C 
regarded as the bounding curve of S, and the positive normal of S 
are so oriented to each other as the positive axes of x, y, and z 
respectively. 
An Invariant Property. It will be shown in Chap. XIII that, if 
the coordinates are transformed to any new system of Cartesian axes, 
provided merely that a right-handed system does not go into a left- 
handed system, the integrands of both the line integral and the sur 
face integral in I. will preserve their form. Thus 
,, dx' 
d\ 
■ Q'dy_ jp dz _ jj dx 
r ^ rl\ ' rl\ 
+ 0^+72* 
and similarly, the integrand of the transformed surface integral 
will be * 
2£\ 
dz'J 
cos a' -f- 
dP' 
dz' 
cR'\ 
ex' J 
cos /3' +(°Jrj 
\cx 
dP'\ 
dy'J 
COS y\ 
But Equation I. is invariant even of a reflection, as z' = — z, 
if E' = — R-, for then y' = tt — y and the sense of C is reversed. 
Suppose now that we have an arbitrary bounded surface, S. Then 
we can cut it up into a finite number of pieces, Si, S 2 , •••, each of 
which, referred to a system of Cartesian axes properly chosen, will 
come under the case just treated. Hence Stokes’s theorem will hold 
for such a piece, no matter how the axes are chosen, and so we may 
refer all the pieces to the same axes. 
Write down, then, Stokes’s theorem for each of the pieces Si, S 2 , •••, 
and add the results together. For the kind of surfaces we most 
readily think of, like a piece of a sphere or a paraboloid, we shall be 
integrating along each of the cuts once in one direction and once in 
the opposite direction. So these contributions to the sum on the 
* These facts can, however, be proved here directly by the student, by merely 
writing down the most general equations which represent such a transformation 
(.Analytic Geometry, p. 592 and p. 594), and then computing the original inte 
grands in terms of the new variables.