246 
CALCULUS 
When At approaches 0, the integral approaches the integral of the 
limiting function, as will be shown under the proof of Leibniz’s 
Rule. Thus 
V 
We can, however, assemble all our physical hypotheses into the 
single one : 
Physical Hypothesis. If in the case of a flow of heat the temper 
ature is a continuous function of x, y, z, t, and if this is true of du/dt, 
too, then the rate at which the heat is accumulating in a given region, V, 
is given by the formula : 
(11) 
V 
where the specific heat, C, is either a constant or a contimious function 
of x, y, z, t. 
Prom (11) equations (9) and (10) follow at once. 
14. The Heat Equation. Consider an arbitrary flow of heat, in 
which the temperature, together with the partial derivatives * of 
the first two orders, is continuous in x, y, z, t. Let V be an arbi 
trary sub-region contained in the region of flow. Then the rate at 
which the heat in V is increasing is given in two forms, namely, by 
equation (8) of § 12 and by (11) of § 13. Hence 
(12) 
where n refers to the inner normal of S. For ordinary substances 
and moderate variations in the temperature, K may be assumed 
constant. 
By Green’s Theorem, § 9, we have: 
Hence 
* It is not necessary to extend this requirement to all these derivatives ; but 
the loss in generality is unimportant.