\[\mathbf{r}=\mathbf(\alpha , \beta )\]
, a normal vector is \[\mathbf{N} =\frac{\partial \mathbf{r}}{\partial \alpha} \times \frac{\partial \mathbf{r}}{\partial \beta} \]
.To see this, at a point
\[(\alpha_0 , \beta_0 )\]
in the surface, \[\frac{\partial \mathbf{r}}{\partial \alpha}_{(\alpha_0 , \beta_0 )}\]
and \[\frac{\partial \mathbf{r}}{\partial \beta}_{(\alpha_0 , \beta_0 )}\]
are both tangential to the surface so \[\mathbf{N} =\frac{\partial \mathbf{r}}{\partial \alpha}|_{(\alpha_0 , \beta_0 )} \times \frac{\partial \mathbf{r}}{\partial \beta}|_{(\alpha_0 , \beta_0 )} \]
is normal to the surface.