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Unformatted text preview: tionality being determined by the density and
speciﬁc heat of the material making up the bar. More generally, if σ (x) denotes
the speciﬁc heat at the point x and ρ(x) is the density of the bar at x, then the
heat within the region Dx1 ,x2 between x1 and x2 is given by the formula
x2 Heat within Dx1 ,x2 = ρ(x)σ (x)u(x, t)dx.
x1 To calculate the rate of change of heat within Dx1 ,x2 with respect to time, we
simply diﬀerentiate under the integral sign:
x2 d
dt x2 ρ(x)σ (x)u(x, t)dx = ρ(x)σ (x) x1 x1 ∂u
(x, t)dx.
∂t Now heat is a form of energy, and conservation of energy implies that if no
heat is being created or destroyed in Dx1 ,x2 , the rate of change of heat within
Dx1 ,x2 is simply the rate at which heat enters Dx1 ,x2 . Hence the rate at which
heat leaves Dx1 ,x2 is given by the exp
x2 Rate at which heat leaves Dx1 ,x2 = − ρ(x)σ (x)
x1 ∂u
(x, t)dx.
∂t (4.1) (More generally, if heat is being created within Dx1 ,x2 , say by a chemical reaction, at the rate µ(x)u(x, t) + ν (x) per...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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