$2\mathrm{.}8.$ Determine the Euler equation and natural boundary conditions obtainable from the minimization of the functional:

$I(T)={\displaystyle \underset{}{\overset{\phantom{}}{}}}\{\frac{1}{2}[grad(T)*k*grad(T)]+QT\}dxdy-{\displaystyle \underset{S_1}{\overset{\phantom{}}{}}}{q}^{\text{'}\text{'}}Tds+{\displaystyle \underset{S_2}{\overset{\phantom{}}{}}}h(\frac{1}{2}T-T_{})ds$

where T is the temperature, $k$ is the matrix of thermal conductivity assuming that this thermophysical property has the form $k_{ij},$ with $i,j=1,2,3$ for the three coordinate directions $(x,y,z).Q$ is the volumetric heat generation rate, $q^{\text{'}\text{'}}$ is the magnitude of the heat flux vector, $h$ is the heat transfer coefficient, and $T_{}$ is the temperature of the ambient in which the $($solid$)$ domain $$ is located, while $S_1$ and $S_2$ are two regions on the surface of $$

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