Heat Flow in a Material of Variable Thermal Conductivity With Internal Heat Source

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  
\[ \mathbf{\nabla} \cdot (k \mathbf{\nabla} T) = - \phi\]

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