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5.2 Three-Dimensional Heat Equation 299

Figure 6 The heat flow rate through a small section of surface with area Ais
q·nˆ A.

similar units. Then the rate at which heat is generated in a small region of vol-
umeVcentered on pointPis approximatelyg(P,t) V. These contributions
are summed over the whole subregionV;as Vshrinks, their total becomes
the integral
∫∫∫

V

g(P,t)dV.

The rate at which heat is stored in a small region of volume Vcentered on
pointPis proportional to the rate at which temperature changes there. That
is, the storage rate isρc Vut(P,t). The storage rate for the whole subregion
Vis the sum of such contributions, which passes to the integral
∫∫∫

V

ρc∂∂ut(P,t)dV.

Now the heat balance equation in mathematical terms becomes

∫∫

S

−q·nˆdA+

∫∫∫

V

gdV=

∫∫∫

V

ρc

∂u

∂tdV. (1)

At this point, we call on the divergence theorem, which states that the integral
over a surfaceSof the outward normal component of a vector function equals
the integral over the volume bounded bySof the divergence of the function.
Thus
∫∫

S

q·nˆdA=

∫∫∫

V

∇·qdV (2)

and we make this replacement in Eq. (1). Next collect all terms on one side of
the equation to find
∫∫∫

V

[

−∇ ·q+g−ρc

∂u

∂t

]

dV= 0. (3)