PlanetPhysics/Derivation of Heat Equation

Let us consider the heat conduction in a homogeneous matter with density $$\varrho$$ and specific heat capacity $$c$$.\, Denote by \,$$u(x,\,y,\,z,\,t)$$\, the temperature in the point \,$$(x,\,y,\,z)$$\, at the time $$t$$.\, Let $$a$$ be a simple closed surface in the matter and $$v$$ the spatial region restricted by it.

When the growth of the temperature of a volume element $$dv$$ in the time $$dt$$ is $$du$$, the element releases the amount $$-du\;c\,\varrho\,dv \;=\; -u'_t\,dt\,c\,\varrho\,dv$$ of heat, which is the heat flux through the surface of $$dv$$.\, Thus if there are no sources and sinks of heat in $$v$$, the heat flux through the surface $$a$$ in $$dt$$ is $$\begin{matrix} -dt\int_vc\varrho u'_t\,dv. \end{matrix}$$ On the other hand, the flux through $$da$$ in the time $$dt$$ must be proportional to $$a$$, to $$dt$$ and to the derivative of the temperature in the direction of the normal line of the surface element $$da$$, i.e. the flux is $$-k\,\nabla{u}\cdot d\vec{a}\;dt,$$ where $$k$$ is a positive constant (because the heat flows always from higher temperature to lower one).\, Consequently, the heat flux through the whole surface $$a$$ is $$-dt\oint_ak\nabla{u}\cdot d\vec{a},$$ which is, by the Gauss's theorem, same as $$\begin{matrix} -dt\int_vk\,\nabla\cdot\nabla{u}\,dv \;=\; -dt\int_vk\,\nabla^2u\,dv. \end{matrix}$$ Equating the expressions (1) and (2) and dividing by $$dy$$, one obtains $$\int_vk\,\nabla^2u\,dv \;=\; \int_vc\,\varrho u'_t\,dv.$$ Since this equation is valid for any region $$v$$ in the matter, we infer that $$k\,\nabla^2u \;=\; c\,\varrho u'_t.$$ Denoting\, $$\frac{k}{c\varrho} = \alpha^2$$,\, we can write this equation as $$\begin{matrix} \alpha^2\nabla^2u \;=\; \frac{\partial u}{\partial t}. \end{matrix}$$ This is the differential equation of heat conduction, first derived by Fourier.