# we see that bk sinhkba is the k th coecient in the

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Unformatted text preview: olutions. 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 − d dt ρσudxdydz = − D ρσ D by diﬀerentiating under the integral sign. 116 ∂u dxdydz, ∂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.

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