The surface-area element $\mathrm{d\σ}$ is actually the area of an "infinitesimal" parallelogram attached to the surface and formed by to "infinitesimal" tangent vectors along two intersecting coordinate curves. If the surface is represented via the position vector R, then ${\mathbf{R}}_x \mathrm{dx}$ and ${\mathbf{R}}_y \mathrm{dy}$ will be infinitesimal tangent vectors along the coordinate curves $\mathbf{R}\left(x\,a\right)$ and $\mathbf{R}\left(b\,y\right)$, respectively. The cross product of these tangent vectors is normal to the surface, and its length is the area of the infinitesimal parallelogram formed by these tangent vectors at $\left(a\,b\right)$. Table 6.3.14(a) summarizes these relationships.$$

The surface-area element $\mathrm{d\σ}$ is the length of N, so $\mathrm{d\σ}\=∥\mathbf{N}∥$ = $\sqrt{1\+f_x^2\+f_y^2} \mathrm{dA}$.

Since $\mathrm{d\σ}\=\mathrm{\λ} \mathrm{dA}$, the calculations in Table 6.3.14(b) lead to $\mathrm{\λ}\=\sqrt{1\+f_x^2\+f_y^2}$. In the Cartesian coordinates assumed in Table 6.3.2, $\mathrm{dA}$ will have the form $\mathrm{dy} \mathrm{dx}$ or $\mathrm{dx} \mathrm{dy}$. The surface-area element is actually the area of an "infinitesimal" parallelogram attached to the surface and formed by to "infinitesimal" tangent vectors along two intersecting coordinate curves. The difference between an infinitesimal tangent vector and a finite one is just the differential factor (either $\mathrm{dx}$ or $\mathrm{dy}$). Hence, the calculations in Table 6.3.14(b) omit these differentials, and lead to $\mathrm{\λ}$ rather than $\mathrm{d\σ}$.