Unformatted text preview: unit volume, then the rate at which heat
leaves Dx1 ,x2 is
− x2 ρ(x)σ (x)
x1 ∂u
(x, t)dx +
∂t x2 (µ(x)u(x, t) + ν (x))dx.)
x1 On the other hand, the rate of heat ﬂow F (x, t) is proportional to the partial
derivative of temperature,
F (x, t) = −κ(x) ∂u
(x, t),
∂x (4.2) where κ(x) is the thermal conductivity of the bar at x. Thus we ﬁnd that the
rate at which heat leaves the region Dx1 ,x2 is also given by the formula
F (x2 , t) − F (x1 , t) = x2
x1 ∂F
(x, t)dx.
∂x (4.3) 1 Further reading can be found in the many excellent upperdivision texts on partial diﬀerential equations. We especially recommend Mark Pinsky, Partial diﬀerential equations and
boundaryvalue problems with applications , 2nd edition, McGrawHill, 1991. An excellent
but much more advanced book is Michael Taylor, Partial diﬀerential equations: basic theory ,
Springer, New York, 1996. 82 Comparing the two formulae (4.1) and (4.3), we ﬁnd that
x2
x1 ∂F
(x, t)dx = −
∂x x2 ρ(x)σ (x)
x1 ∂u
(x, t)dx.
∂t This equation is true for all choices of x1 and x2 , so the in...
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 Winter '14
 Equations

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