Solution
= Solution
The area element is $\sqrt{\det I^S}\,dx\,dy=W\,dx\,dy$, so
$$
\boxed{A(h)=\int_\Omega\sqrt{1+h_x^2+h_y^2}\,dx\,dy.}
$$
For the <Lagrangian> $L(p,q)=\sqrt{1+p^2+q^2}$, stationarity under every compactly supported variation gives the <Euler-Lagrange equation>
$$
\partial_x\!\left(\frac{h_x}{W}\right)
+\partial_y\!\left(\frac{h_y}{W}\right)=0.
$$
Multiplication by $W^3$ expands this into
$$
(1+h_y^2)h_{xx}-2h_xh_yh_{xy}+(1+h_x^2)h_{yy}=0,
$$
recovering part (d).