Surface area

The area $s$ of the parallelogram defined by vectors $\vec{v}$ and $\vec{w}$ is:

$$s = \vert\vec{v}\times\vec{w}\vert$$

Considering a parameterization and a small patch $R$ bounded in $u$ by $u_0$ and $u_0 + d_u$ and in $v$ by $v_0$ and $v_0 + d_v$, assuming $d_u$ and $d_v$ are positive:

$$\begin{array}{rcl} ds&=&\vert d_u x_u\times d_v x_v\vert\\ ... ... (x_u\cdot x_v)^2}\\ &=&d_u d_v\sqrt{EG - F^2}\\ \end{array}$$

Note that $EG - F^2$ is the determinent of the first fundamental matrix. Thus the area:

$$A = \int\int_W \sqrt{EG - F^2}d_u d_v$$

where $W$ is the set of points in the parameter space that map onto $R$. It can be shown that the area of a surface is parameterization-independent.