Page 425 - Advanced engineering mathematics

12.8 The Divergence Theorem of Gauss 405

z

y

k

x
n
Σ j

M

FIGURE 12.22 Archimedes’s
Principle.

ρzn · kA j . Sum these vertical components over to obtain approximately the buoyant force on
the object, then take the limit as the surface elements are chosen smaller. We obtain

net buoyant force on = ρzn · kdσ.

Write this integral as ρzk · ndσ and apply the divergence theorem to obtain

net buoyant force on = ∇· (ρzk)dV.
M
But ∇· (ρzk) = ρ,so

net buoyant force on = ρ dV = ρ[volume of M].
M
and ρ multiplied by the volume of M is the weight of the object.

12.8.2 The Heat Equation
We will use the divergence theorem to derive a partial differential equation that models heat
conduction and diffusion processes. Suppose some medium (such as a bar of metal or water in
a pool) has density ρ(x, y, z), specific heat μ(x, y, z) and coefficient of thermal conductivity
K(x, y, z).Let u(x, y, z,t) be the temperature of the medium at (x, y, z) and time t. We want an
equation for u.
We will exploit an idea frequently used in constructing mathematical models. Consider an
imaginary smooth closed surface in the medium, bounding a solid region M. The amount of
heat energy leaving M across in a time interval t is

(K∇u) · ndσ t.

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October 14, 2010 14:53 THM/NEIL Page-405 27410_12_ch12_p367-424