Solution
= Solution
Parameterize the surface by $\mathbf r(x,z)=(x,1+z,z)^T$. Its two <tangent vectors> are $(1,0,0)^T$ and $(0,1,1)^T$, whose <cross product> is $(0,-1,1)^T$ of <Euclidean norm> $\sqrt2$. Thus the scalar <area element of a surface> is
$$
\boxed{dS=\sqrt2\,dx\,dz}.
$$
Equivalently, projection onto the $xz$-<plane> reduces the surface <area> by $|\mathbf n\cdot\mathbf e_y|=1/\sqrt2$.