# Note that the heat equation itself and the boundary

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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 upper-division texts on partial diﬀerential equations. We especially recommend Mark Pinsky, Partial diﬀerential equations and boundary-value problems with applications , 2nd edition, McGraw-Hill, 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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