LINE INTEGRALS AND GREEN’S THEOREM 241 
dp_dQ = o 
dy CZ ’ CZ dx ’ cx 'ey 
Consider the integral 
/ Pclx + Q dy + R dz. 
C 
Its value will be the same for all intermediate positions of C. For, 
two near positions form the complete boundary of a bi-lateral sur 
face contained in V, and hence this integral, extended over both 
positions of C (but in opposite senses) vanishes by Stokes’s Theorem. 
Let M be a diametral plane of the sphere K. Then C\ can be 
deformed continuously toward its projection on M. Again, the 
value of the integral remains constant. The limiting position of 
M is, however, a closed curve Y (no longer simple, in general) which 
lies in M. But for a closed plane curve, simple or not, the integral 
vanishes by § 5. Hence the original integral = 0. 
Finally, the integral 
(x, y, z) 
Pdx 4- Qdy + Rdz 
has the same value for all paths connecting (a, b, c) with (x, y, z) 
and lying in V, and hence it defines a single-valued function, u, in V. 
Thus all the theorems of § 8 are established. Stokes’s Theorem 
owes its importance, however, chiefly to those cases in physics, in 
which the surface integral has a meaning. 
11. Flow of Heat. Imagine a slab of copper 2 cm. thick, with 
one side packed in melting ice at temperature u = u 0 = 0°, and the 
other side exposed to steam, 
are permanently maintained, p IG . 59 
the flow will tend toward a 
limiting condition, in which the lines of flow are the perpendiculars 
to the faces of the slab, and the isothermal surfaces are the planes 
parallel to these faces. Moreover, the temperature will fall off 
steadily, as a point P traces a line of flow. If x denotes the dis 
tance of P from the surface of temperature u l} and a, the thickness 
of the plate, then 
(1) u = Ui -(if! — Mo)- • 
a