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In the rest of this section, we consider our ﬁrst example, the equation governing heat ﬂow through a region of (x, y, z )-space, under the assumption that
no heat is being created or destroyed within the region. Let
u(x, y, z, t) = temperature at (x, y, z ) at time t.
If σ (x, y, z ) is the speciﬁc heat at the point (x, y, z ) and ρ(x, y, z ) is the
density of the medium at (x, y, z ), then the heat within a given region D in
(x, y, z )-space is given by the formula
Heat within D = ρ(x, y, z )σ (x, y, z )u(x, y, z, t)dxdydz.
D If no heat is being created or destroyed within D, then by conservation of energy,
the rate at which heat leaves D equals minus the rate of change of heat within
D, which is
dt ρσudxdydz = −
D by diﬀerentiating under the integral sign.
∂t (5.1) On the other hand, heat ﬂow can be represented by a vector ﬁeld F(x, y, z, t)
which points in the direction of greatest decrease of temperature,
F(x, y, z, t) = −κ(x, y, z )(∇u)(x, y, z, t),
where κ(x, y, z ) is the so-called thermal conductivity of the medium at (x, y, z ).
Thus the rate at which heat leaves the regi...
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This document was uploaded on 01/12/2014.
- Winter '14