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Unformatted text preview: on D through a small region in its
boundary of area dA is
−(κ∇u) · NdA,
where N is the unit normal which points out of D. The total rate at which heat
leaves D is given by the ﬂux integral
− (κ∇u) · NdA,
∂D where ∂D is the surface bounding D. It follows from the divergence theorem
that
Rate at which heat leaves D = − ∇ · (κ∇u)dxdydz. (5.2) D From formulae (5.1) and (5.2), we conclude that
ρσ
D ∂u
dxdydz =
∂t ∇ · (κ∇u)dxdydz.
D This equation is true for all choices of the region D, so the integrands on the
two sides must be equal:
ρ(x, y, z )σ (x, y, z ) ∂u
(x, y, z, t) = ∇ · (κ∇u)(x, y, z, t).
∂t Thus we ﬁnally obtain the heat equation
∂u
1
=
∇ · (κ(x, y, z )(∇u)) .
∂t
ρ(x, y, z )σ (x, y, z )
In the special case where the region D is homogeneous , i.e. its properties are
the same at every point, ρ(x, y, z ), σ (x, y, z ) and κ(x, y, z ) are constants, and
the heat equation becomes
∂u
κ ∂2u ∂2u ∂2u
=
+ 2+ 2 .
∂t
ρσ ∂x2
∂y...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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