Derivation of the Fundamental Equation of Heat Conduction

The Fundamental Heat Flow equation is  
\[c \rho \frac{\partial T}{\partial t} - k \nabla^2 T=0\]
, where
  is thew specific heat capacity
  is the density of the conducting material
  is the thermal conductivity of the material.
If the flow of heat is represented by a vector  
  then the rate of heat flow through a surface is equal to  
\[\int \int_S \mathbf{u} \cdot \mathbf{n} dS\]

By the Divergence Theorem  
\[\int \int_S \mathbf{u} \cdot \mathbf{n} dS = \int \int \int_V \mathbf{\nabla} \cdot \mathbf{u}dV\]

The basic law of heat conduction is that  
\[\mathbf{u}= - k \mathbf{\nabla} T \]

\[\int \int_S - k ( \mathbf{\nabla} T) \cdot \mathbf{n} dS = -k \int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} T) dV\]

The rate at which heat is being gained per unit mass from the material is  
\[c \frac{\partial T}{\partial t}\]
  and the rate at which heat is bring gained by the region  
\[\int \int \int_V \frac{\partial T}{\partial t} dV\]

\[ k \int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} T) dV = \int \int \int_V \frac{\partial T}{\partial t} dV\]

\[ \int \int \int_V (k \mathbf{\nabla} \cdot (\mathbf{\nabla} T) - \frac{\partial T}{\partial t} )dV=0\]

\[ ( k \mathbf{\nabla} \cdot (\mathbf{\nabla} T) - \frac{\partial T}{\partial t} )=0\]

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