LINE INTEGRALS AND GREEN’S THEOREM 247 
The integrand of this last integral is continuous throughout the 
whole region of flow. Hence it must vanish at every point of the 
region : 
dt 
For, if it were, for example, positive at a point P within the region, 
it would be positive throughout a certain neighborhood of P. But 
the integral of a positive function cannot be zero. 
We thus arrive at the differential equation which governs the 
flow of heat in the general case: 
(13) 
It is a linear partial differential equation of the second order with 
constant coefficients. 
Steady Flow. We can now define a steady flow as one in which 
the temperature at any given point is independent of the time. 
From (13) it follows that the temperature, in the case of a steady 
flow, will satisfy Laplace’s Equation: 
Conversely, if the temperature satisfies Laplace’s Equation, then 
from (13) du/dt = 0, and the flow is steady. 
A necessary and sufficient condition for a steady flow is the fol 
lowing : if V be an arbitrary sub-region contained in the region of 
flow, then cQ/dt = 0 for this region. 
The latter property might be taken as the definition of a steady 
flow. 
EXERCISE 
If K is variable, but continuous, together with its partial deriva 
tives of the first order, show that the heat equation becomes : 
du 1 
Tt~c 
Here, C may also be variable ; it will be continuous. 
15. Flow of Electricity in Conductors. The flow of electricity in 
a conductor is mathematically identical with the problem of the flow 
of heat just discussed. On replacing throughout the word heat by 
electricity, and the word temperature by potential, the foregoing treat 
ment applies to the electrical case.