Suppose in a material of variable thermal conductivity
\[k=k(x,y,z)\]
the termperature varies from point to point so that \[T=T(x,y,z)\]
Suppose also that a source inside the material generates heat at a constant rate
\[\phi\]
Joules per unit volume.Heat flows at a rate
\[\mathbf{h}\]
per unit area perpendicular to a surface drawn in the material. then \[\mathbf{\nabla} \cdot \mathbf{h} = loss \: of \: heat \: per \: unit \: volume\]
For the temperature to remain steady
\[\mathbf{\nabla} \cdot \mathbf{h} = \phi \]
But
\[ \mathbf{h} = -k \mathbf{\nabla} T\]
Hence
\[ \mathbf{\nabla} \cdot (k \mathbf{\nabla} T) = - \phi\]