Structural Column Strain Energy Stability (Civil & Mechanical Engineering)

A load-bearing structural steel support column in a commercial building framework experiences internal strain energy $U$ (in Joules, $\text{J}$) due to combined axial mechanical compression loads and solar thermal expansion. The total strain energy is modeled as a function of the axial force load $x$ (in kilonewtons, $\text{kN}$) and the surface temperature shift $y$ (in degrees Celsius, $^\circ\text{C}$):

1. Compute the first-order partial derivatives $U_x$ and $U_y$.
2. Compute all four second-order partial derivatives ($U_{xx}, U_{yy}, U_{xy}, U_{yx}$).
3. Verify Clairaut's Theorem for the mixed partial derivatives.
4. Construct the $2 \times 2$ Hessian matrix $\mathbf{H}_U(x, y)$ and evaluate its determinant $D = \det(\mathbf{H}_U)$ at nominal service load $(x, y) = (10\text{ kN}, 25^\circ\text{C})$.
5. Interpret the physical structural meaning of the mixed partial derivative $U_{xy}$ and explain why $U_{xx} > 0$ confirms structural stability against buckling.