The Fundamental Heat Flow equation is  {jatex options:inline}c \rho \frac{\partial T}{\partial t} - k \nabla^2 T=0{/jatex}, where
{jatex options:inline}c{/jatex}  is thew specific heat capacity
{jatex options:inline}\rho{/jatex}  is the density of the conducting material
{jatex options:inline}k{/jatex}  is the thermal conductivity of the material.
If the flow of heat is represented by a vector  {jatex options:inline}\mathbf{u}{/jatex}  then the rate of heat flow through a surface is equal to  {jatex options:inline}\int \int_S \mathbf{u} \cdot \mathbf{n} dS{/jatex}
By the Divergence Theorem  {jatex options:inline}\int \int_S \mathbf{u} \cdot \mathbf{n} dS = \int \int \int_V \mathbf{\nabla} \cdot \mathbf{u}dV{/jatex}
The basic law of heat conduction is that  {jatex options:inline}\mathbf{u}= - k \mathbf{\nabla} T {/jatex}
Hence  {jatex options:inline}\int \int_S - k ( \mathbf{\nabla} T) \cdot \mathbf{n} dS = -k \int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} T) dV{/jatex}
The rate at which heat is being gained per unit mass from the material is  {jatex options:inline}c \frac{\partial T}{\partial t}{/jatex}  and the rate at which heat is bring gained by the region  {jatex options:inline}V{/jatex}  is  {jatex options:inline}\int \int \int_V \frac{\partial T}{\partial t} dV{/jatex}
Hence  {jatex options:inline} k \int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} T) dV = \int \int \int_V \frac{\partial T}{\partial t} dV{/jatex}
Then  {jatex options:inline} \int \int \int_V (k \mathbf{\nabla} \cdot (\mathbf{\nabla} T) - \frac{\partial T}{\partial t} )dV=0{/jatex}
Finally  {jatex options:inline} ( k \mathbf{\nabla} \cdot (\mathbf{\nabla} T) - \frac{\partial T}{\partial t} )=0{/jatex}