derivation of heat equation
Let us consider the heat
conduction
in a homogeneous matter with density
and specific heat capacity
. Denote by
the temperature
in the point
at the time
. Let
be a simple closed surface in the matter and
the spatial region restricted by it.
When the growth of the temperature of a volume
element
in the time
is
, the element releases the amount
of heat, which is the heat flux through the surface of
. Thus if there are no sources and sinks of heat in
, the heat flux through the surface
in
is
 |
(1) |
On the other hand, the flux through
in the time
must be proportional to
, to
and to the derivative of the temperature in the direction of the normal line of the surface element
, i.e. the flux is
where
is a positive constant (because the heat flows always from higher temperature to lower one). Consequently, the heat flux through the whole surface
is
which is, by the Gauss's theorem, same as
 |
(2) |
Equating the expressions (1) and (2) and dividing by
, one obtains
Since this equation is valid for any region
in the matter, we infer that
Denoting
, we can write this equation as
 |
(3) |
This is the differential equation of heat conduction, first derived by Fourier.
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