surface area (parametric form)

The true curved area of a parametric surface, found by weighting the flat parameter domain by the stretch factor $|\mathbf{r}_u\times\mathbf{r}_v|$ .

A flat rectangle of sides $\Delta u$ and $\Delta v$ in the $uv$ -plane maps to a small, possibly stretched or squashed curved patch on the surface. The normal vector's own length, $|\mathbf{r}_u\times\mathbf{r}_v|$ , is the area of the parallelogram spanned by $\mathbf{r}_u$ and $\mathbf{r}_v$ — the stretch factor. Surface area is $S=\iint_D |\mathbf{r}_u\times\mathbf{r}_v|\,du\,dv$ . For the graph special case, this reduces to the familiar $\sqrt{f_x^2+f_y^2+1}$ integrand.

Examples

For the cone $\mathbf{r}(u,v)=\langle u\cos v,\,u\sin v,\,u\rangle$ , $0\le u\le2$ : $S = \int_0^{2\pi}\int_0^2 u\sqrt2\;du\,dv = 4\sqrt2\,\pi \approx 17.77$ .